Can an Eastern rite priest concelebrate at a western rite Mass and vice versa?

The 1983 Code of Canon Law for the Latin rite is silent regarding this subject, but it was specifically addressed in the Code of Canons of the Eastern Churches

Canon 701 of this code states: "For a just cause and with the permission of the eparchial bishop, bishops and presbyters of different churches 'sui iuris' can concelebrate, especially to foster love and to manifest the unity of the Churches. All follow the prescripts of the liturgical books of the principal celebrant, avoiding any liturgical syncretism whatever, and preferably with all wearing the liturgical vestments and insignia of their own Church 'sui iuris.'"

As it is the only norm available, its indications are also applicable to Roman-rite priests.

To this may be added the injunction of the instruction "Redemptionis Sacramentum," No. 113: "Where it happens that some of the Priests who are present do not know the language of the celebration and therefore are not capable of pronouncing the parts of the Eucharistic Prayer proper to them, they should not concelebrate, but instead should attend the celebration in choral dress in accordance with the norms."

Therefore a priest, with sufficient dominion of the language, may receive permission to concelebrate in a Mass of a rite different from his own. This permission may be for a specific situation or habitual, as may be the case of some Eastern priests studying or residing in Latin-rite institutes or priestly residences.

The priest should normally participate in the vestments of his own rite, as was clearly visible in Pope John Paul II's funeral and Benedict XVI's installation Mass where each of the Eastern cardinals who concelebrated wore his own proper vestments.

Canon 7—2 of the Eastern code, however, allows for exceptions to be made to this rule if proper vestments are unavailable. This could easily happen to an Eastern priest traveling in Europe or America or, less likely, a Latin-rite priest in an Eastern country. In such cases he may concelebrate using Latin or Eastern vestments.

As the canon says, there should be no attempt to syncretize or mix elements of different rites. Only one rite may be followed.

The texts and rubrics of some Eastern rites have been translated into several languages. In some cases these translations are for study purposes only, and priests may never concelebrate using a translation that has not been formally approved for liturgical use by the proper authorities.