Pope Innocent III taught (what is now called) the Byzantine Sign of the Cross as the traditional practice of the Latin Church. He did not speak negatively about the innovators changing it from left to right.I disagree.
A common posture, to be observed by all participants, is a sign of the unity of the members of the Christian community gathered for the sacred Liturgy: it both expresses and fosters the intention and spiritual attitude of the participants.GIRM 42.(Emphasis added).
- The gestures and posture of the priest, the deacon, and the ministers, as well as those of the people, ought to contribute to making the entire celebration resplendent with beauty and noble simplicity, so that the true and full meaning of the different parts of the celebration is evident and that the participation of all is fostered.[52] Therefore, attention should be paid to what is determined by this General Instruction and the traditional practice of the Roman Rite and to what serves the common spiritual good of the People of God, rather than private inclination or arbitrary choice.
In some ways, this is common sense. I wouldn’t expect it to be permissible to use the Latin way of using the sign of the cross in the Melkite Church.
"The sign of the cross is made with three fingers, because the signing is done together with the invocation of the Trinity. … This is how it is done: from above to below, and from the right to the left, because Christ descended from the heavens to the earth, and from the Jews (right) He passed to the Gentiles (left).
“Others, however, make the sign of the cross from the left to the right, because from misery (left) we must cross over to glory (right), just as Christ crossed over from death to life, and from Hades to Paradise. [Some priests] do it this way so that they and the people will be signing themselves in the same way. You can easily verify this — picture the priest facing the people for the blessing — when we make the sign of the cross over the people, it is from left to right…”
- Pope Innocent III (1198-1216)