Word for equivalence preserving transformations of equations
Though I agree this question will likely find its answer more easily on Mathematics SE, I'll try to answer briefly here, and forewarn you of the rather picky tendencies of mathematical language.
The best I can offer is a Homomorphism (Wikipedia), which is a term from abstract algebra, but it sounds as thought it might fit your criteria.
PS: You assert in a comment that "equivalence" is a well-defined mathematical concept, but that is only true in-context. See the Mathematics section of Wikipedia's Equivalence - Disambiguation, for example. Equivalence can mean much more than whether two sides of an equation produce the same result.
I am not aware of a similar word in English, but I believe the terms you are looking for are as follows:
Addition Property of Equality: If a, b, and c are numbers and a = b, then a + c = b + c.
Multiplication Property of Equality: If a, b, and c are numbers and a = b, then ac = bc.
A few technicalities aside, in general you can apply the same function f(x) to both sides of any equation: if a = b, then f(a) = f(b).
So maybe you are looking for Function Property of Equality. Unfortunately not one word. NOTE: https://en.wikipedia.org/wiki/Equality_(mathematics) refers to this as the Substitution Property of Equality.
By the way, you can square both sides of an equation while maintaining equality: if a = b, then a2 = b2.
"Equivalent transformation" is acceptable. https://hal.archives-ouvertes.fr/hal-02416421/document