Actually, I do remember reading in the Vespers doxasticon for the Exaltation the following:
Come, all ye peoples, and let us venerate the blessed Wood, through which the eternal
justice has been brought to pass. For he who by a tree deceived our forefather
Adam, is by the Cross himself deceived; and he who by tyranny gained
possession of the creature endowed by God with royal dignity, is overthrown
in headlong fall. By the Blood of God the poison of the serpent is washed
away; and the curse of a just condemnation is loosed by the unjust
punishment inflicted on the Just. For it was fitting that wood should be healed
by wood, and that through the Passion of One Who knew not passion should
be remitted all the sufferings of him who was condemned because of wood.
But glory to Thee, O Christ our King, for Thy dread dispensation towards us,
whereby Thou hast saved us all, for Thou art good and lovest mankind.
Δεῦτε ἅπαντα τὰ ἔθνη, τὸ εὐλογημένον ξύλον προσκυνήσωμεν, δι᾿ οὗ γέγονεν ἡ αἰώνιος δικαιοσύνη· τὸν γὰρ Προπάτορα Ἀδάμ, ὁ ἀπατήσας ἐν ξύλῳ, τῷ Σταυρῷ δελεάζεται· καὶ πίστει κατενεχθεὶς πτῶμα ἐξαίσιον, ὁ τυραννίδι κρατήσας τοῦ βασιλείου πλάσματος, Αἵματι Θεοῦ, ὁ ἰὸς τοῦ ὄφεως ἀποπλύνεται· καὶ κατάρα λέλυται, καταδίκης δικαίας, ἀδίκῳ δίκῃ τοῦ δικαίου κατακριθέντος· ξύλῳ γὰρ ἔδει τὸ ξύλον ἰάσασθαι, καὶ πάθει τοῦ ἀπαθοῦς, τὰ ἐν ξύλῳ λῦσαι πάθη τοῦ κατακρίτου. Ἀλλὰ δόξα Χριστὲ Βασιλεῦ, τῇ περὶ ἡμᾶς σου φρικτῇ oικονομίᾳ δι᾿ ἧς ἔσωσας πάντας, ὡς ἀγαθὸς καὶ φιλάνθρωπος.
So language of "punishment" or "penalty", and also "justice", seems to be acceptable in Orthodoxy. My impression is that the problem with Western theories of atonement (what we call "penal substitution") is that, following Anselm, they assumed that in some sense Christ became juridically guilty for the sake of His Sacrifice, therefore "earning" vicariously the penalty that the rest of us deserved before. I don't know exactly how Anselm argues for this, but perhaps because if Christ remained completely blameless, even as He suffered death for us on the Cross, one might accuse God the Father of being unjust.
In any case, I'm pretty sure this, at least, is not Orthodox. As the hymn demonstrates, Christ remained completely blameless even as He sacrificed Himself for us on the Cross. Yes, a penalty needed to be paid in accordance with justice, but through the great mystery of the Cross, it was paid in a manner that allowed both justice and mercy to be satisfied (see below on the right understanding of justice and necessity). If God the Father treated the Son as really guilty, that would in fact be unjust (since He was Just).
NB someone might object that the hymn calls the punishment "unjust" and extrapolate from this that justice played no part in the Sacrifice, but to me this only means that it was unjust insofar as it was inflicted on the Just One. For us, the punishment would have been just.
Here's a nice quote from St Gregory Palamas (Homily 16):
Man was led into his captivity when he experienced God’s wrath, this wrath being the good God’s just abandonment
of man. God had to be reconciled with the human race, for otherwise
mankind could not be set free from the servitude. A sacrifice was needed to
reconcile the Father on high with us and to sanctify us, since we had been
soiled by fellowship with the evil one. There had to be a sacrifice which both
cleansed and was clean, and a purified, sinless priest…. God overturned the
devil through suffering and His Flesh which He offered as a sacrifice to God
the Father, as a pure and altogether holy victim – how great is His gift! – and
reconciled God to the human race.
I think this shows that it's not quite right to say that there was no need to satisfy justice. While it is true that God is not bound by necessity, He is nevertheless just by nature, and so it would be inconsistent with His nature to act unjustly.
In conclusion, I would say "penal satisfaction" is a misnomer: a penalty was required. The "wrong" theory of atonement should be called "penal substitution", since it is indeed wrong to say that Christ actually became guilty for the sake of the Economy. He paid the price for us, yes, but all the while remaining completely innocent.