Trying to understand the two opposing opinions, I found an essay concerning the RC opinion to the council. This selection help show the differences in modern reception of the council.
Canonical Status of the Councilhttp://www.arcaneknowledge.org/catholic/councils/comment06q.htm
Since the ecumenicity of the Quinisext Council has been a matter of dispute, we should review how ecumenical councils had been constituted prior to this synod. The Emperor convoked each ecumenical council, declaring its time, place, and purpose, and invited all the bishops of the Empire. After the partition of the Empire and the barbarian invasions, the Emperor only had jurisdiction over the East. In order for the bishops of the West to be represented, the Emperor would send an invitation to the Pope, who had patriarchal authority over all Latin bishops, and could act on their behalf. If a doctrine of the faith was in dispute, the Pope would prepare a definition of the faith, either on his own initiative or in consultation with the Latin bishops. Ecumenical councils took place in or near the imperial city of Constantinople, so it was generally feasible only for the bishops of that patriarchate to attend. The patriarchs of Alexandria, Antioch, and Jerusalem attended in person or by proxy. The Pope never attended in person, but sent his legates with specific instructions on doctrinal matters.
Once an ecumenical council was convened, the papal legates presided over the synod on behalf of the Pope. The issues at hand were discussed among the bishops, and the pope's definition of faith was read to the assembly. The council would then give its assent to the definition, and use it as a basis for constructing its dogmatic constitutions, which define the faith, determine which persons or writings have fallen into heresy, and prescribe ecclesiastical penalties. The bishops could also issue disciplinary canons. At the close of a general council, the constitutions and canons were submitted to the Emperor so he could promulgate them throughout the Empire, while storing the original documents in the imperial archive. In this way the Emperor acts as guardian of the faith. Since the Pope did not attend in person, and he only authorized his legates to proclaim his definition of the faith, any further acts of the council needed papal ratification. After the council closed, the constitutions and canons were sent to the Pope for his signature. Only with papal approval could the acts be binding in the West, thereby giving the council true ecumenical status. The Pope sometimes elected to exclude or qualify some of the constitutions or canons in his ratification. Any decree or canon excluded by the Pope would not have ecumenical authority, but would at most be binding in the ecclesial jurisdictions of the East.
Since the Council in Trullo did not involve dogmatic matters, but only disciplinary canons, it is understandable why the Greek bishops might not see the need for prior papal approval. Nonetheless, it was highly irregular for an ecumenical council not to be presided by papal legates. The Greeks were evidently aware of this, so Archbishop Basil of Gortyna (in Crete) added to his signature, "Holding the place of the Holy Church of Rome in every synod," just as he had done at the Sixth Ecumenical Council in 680, though there is no evidence he received a perpetual commission from the Pope. The idea that the Trullan council was a continuation of the sixth general synod does not reflect historical reality. As Hefele proved, this council was convened more than a decade later, in 691 or 692. The list of signatories further shows that the Quinisext Council was not constituted of the same bishops who had attended the Sixth Ecumenical Council. This was really a new council, especially since the ecumenical council had long ago been formally closed and ratified.
The Council in Trullo was not ecumenical in its origin, since it was convoked and presided independently of papal authority or any other real participation by the patriarchate of the West, with which the East was not in schism. The claim that this was a continuation of the Sixth Ecumenical Council is untenable, since the general council had been canonically closed and confirmed by the bishops, the Emperor, and the Pope. There is no precedent in Church history, before or since, of reopening such a council, and even if it were possible, this would have to be executed by the same authority that closed the council, and here the authority of the Pope was absent. Therefore, in its origin, the Council in Trullo was a regional synod of the East, notwithstanding its pretensions.
Nonetheless, there is precedent for a regional synod attaining ecumenical status by post facto ratification by the Pope. This was the case with the First Council of Constantinople (381), a synod of Greek bishops that was later ratified by Pope Damasus. Yet in this case the synod was indirectly convoked by the Pope, who urged that the bishops should meet in Rome. Since the Greeks were unable to make the long journey, some of them convened in Constantinople in a parallel synod. At the close of their council, they sent three bishops with a letter to the Pope, "to show that our intentions are peaceful and have unity as their goal."
Such peaceful intentions can hardly be ascribed to some of the Trullan canons, which are overtly hostile to Latin customs. Interestingly, the council's grievances with the West had nothing to do with the authority of the Pope, the filioque, or any of the other major complaints common to later Greek Orthodoxy. The Greeks objected to Latin customs such as priestly celibacy and fasting on Saturday. They did not presume to do away with all non-Greek customs, but only those believed to be in direct conflict with apostolic tradition. Indeed, only about a half dozen of the 102 canons are overtly hostile to Latin customs. In determining apostolic tradition, the Greeks perhaps relied too heavily on the so-called Apostolic Canons, which were mostly written in the fourth century or later. Since these canons were written in the East, they lent credence to the erroneous view that certain Greek liturgical customs were of apostolic origin. Even if these Eastern customs could be traced back to the Apostles, it would not follow that equally ancient customs could not have been established elsewhere in Christendom. The council is marked by an excessive zeal for canonical uniformity, and the modern reader must remember that the distinction between orthodoxy in faith and purely disciplinary matters was not always clearly perceived at this time.
Since we now fully recognize that disciplinary canons are not immutable or irrevocable, the question of the ecumenicity of this council loses some importance. An ecumenical council's grace of infallibility does not properly apply to disciplinary canons, except that they may not incidentally contain anything contrary to divine faith or natural law. Papal ratification of the canons after the fact might raise the canons to ecumenical status in a more ordinary sense, namely that they apply to every jurisdiction in the Church, not that the Council in Trullo is properly ecumenical. Even here the question is somewhat moot, since many of the Trullan canons have fallen into disuse in both the East and the West.
All the patriarchs except the Pope ratified the council. The patriarchs of Alexandria, Antioch, and Jerusalem did not attend the Council in Trullo, but they signed its decrees afterward. The permanent legates who served as papal ambassadors in Constantinople were also induced to sign the decrees, though they had no commission to represent the Pope in a general synod, nor did they attend the council. The place for the Pope's signature, just below that of the Emperor, remained conspicuously blank. Pope Sergius steadfastly refused to sign the acts of the council, claiming he would rather die than endorse its errors. The council was therefore in opposition to the head of the Church, and cannot be considered ecumenical, nor can we invoke the example of the First Council of Constantinople, since no such opposition existed between that council and the Pope. Still, the question remains as to which canons of this local council were ever approved by later popes.
Emperor Justinian later asked Pope John VII (705-07) to ratify the Trullan canons, beseeching the pontiff to convene a Roman synod that would nullify any objectionable canons and confirm the rest. The Pope declined to confirm or condemn any of the canons. His successor, Pope Constantine (708-15), met with the emperor and arrived at some sort of compromise, the terms of which have not been recorded. The fact that the Latins did not change their liturgical customs is sufficient evidence that the Pope did not agree to any of the blatantly objectionable canons, though we cannot be certain regarding what sort of endorsement he may have given to the rest.
The Greeks at the Seventh Ecumenical Council (787) explicitly recognized the Trullan canons as pertaining to the sixth general synod, but this declaration was never ratified by Rome. However, Pope Adrian I (772-795) says he approved the canons of all six councils that were lawfully promulgated, including the canon forbidding depictions of the Precursor (John the Baptist) with the Lamb (as Christ), which is the eighty-second Trullan canon. It seems that Pope Adrian believed that the canons of Trullo pertained to the Sixth Ecumenical Council, yet his approval of the canons was qualified: quae jure ac divinitus ab ipsis promulgatae sunt ("those alone which were lawfully and divinely promulgated"). While the Pope did not challenge the Greek claim that the Trullan canons belonged to the sixth general council, this acquiescence to a historical claim hardly suffices to elevate the council to ecumenical status. Indeed, a century later, Pope John VIII (872-882) referred to the Trullan canons as those which the Greeks "maintain" (perhibent) to be from the sixth general synod, so the question of the council's ecumenicity was by no means considered decided. The Greek claim was based on the belief that the same bishops of the Sixth Ecumenical Council issued the Trullan canons just five years later, but we have seen that this belief is historically inaccurate.
The last definitive papal stance on the Council in Trullo was expressed by the aforementioned John VIII, who accepted from that council "all those canons which did not contradict the true faith, good morals, and decrees of Rome." In particular, the Apostolic See now accepted all eighty-five of the so-called Apostolic Canons, whereas previously it had only accepted the first fifty. Apparently, even at this late date, the Romans had not generally accepted the Trullan canons, which could hardly be the case if Adrian I or any other pope had truly declared the council to be ecumenical.
Relatively few of the Trullan canons were explicitly identified as approved by the Pope. Only the eighty-five Apostolic Canons (mentioned in the second canon of Trullo) are known to have been approved by John VIII, while Adrian I approved at least the eighty-second canon of Trullo. For the remainder of the canons, we are left to apply John VIII's principle of consistency with "the true faith, good morals, and decrees of Rome."
Several canons from the Council in Trullo are included in Gratian's Decretum, but this canonical compilation never had the status of an official code of ecclesiastical law, and we cannot necessarily infer papal approval of a canon from its inclusion in this collection. Nonetheless, the Decretum was a highly influential source of canon law, so it is worth mentioning the canons it cites, as these would not have been considered contrary to the decrees of Rome. Gratian cites the following Trullan canons: the second, fourth, sixth, eleventh, thirteenth, seventeenth, twenty-third, twenty-fourth, twenty-eighth, thirty-sixth, and ninety-third. The thirteenth canon is indicated as being of local application, while many of the other canons are cited only partially or in an oblique paraphrase. Needless to say, nearly all of the Trullan canons were practically ignored in the West.
The Greeks, by contrast, have consistently regarded the Quinisext Council as ecumenical, though even they allowed most of its canons to fall into disuse. In particular, the canons opposed to Latin customs were never used to impose restrictions on Latin liturgy in the East, nor was communion with the Latin Church denied on account of failure to adopt the Trullan canons. Since the schism of 1054, the Orthodox have had little regard for the pronouncements of the Roman Church. However, in the seventh century the Church of the East did not pretend to be the entire Church, and the Greeks recognized the necessity of papal ratification to make a council's acts ecumenical. Thus, even by eastern ecclesiology, the Council in Trullo should not be regarded as ecumenical.