Definition. Two edges e1 and e2 of a graph are called ??? if for any given cycle C they are either both in C or both not in C.
What are such edges called?
The definition obviously defines an equivalence class on the set of edges, what are these classes referred to as?
I'm not aware of terminology exactly describing these, but here is a bit more information about how these edges are related.
First, there is one big equivalence class of this relation: the bridges. These are the edges not on any cycles, and so they are all equivalent. In the intended application of computing the polynomial $$K = \sum_{S} \prod_{e_i \notin S} \tau_i$$ over all spanning trees $S$, these stand out especially in that they do not affect the product at all.
For all other cases, we can look at the slightly stronger equivalence relation: $e_1 \sim e_2$ if (1) these edges are part of at least one cycle together, and (2) every cycle that contains one edge, also contains the other. This $\sim$ is a refinement of a more well-known definition: if we only include condition (1), then the equivalence classes of the edges are exactly the blocks or biconnected components of the graph. So we can look at each block separately to study $\sim$.
(We can also, actually, look at each block separately to study the polynomial $K$. The intersection of a spanning tree of the graph with a block is exactly a spanning tree of the block; conversely, if we take a spanning tree of each block, their union is a spanning tree of the graph. So if we compute similarly-defined polynomials $K_1, K_2, \dots, K_t$ for each block, then $K$ is their product $K_1 K_2 \cdots K_t$.)
Looking at a block means looking at a graph $G$ which is $2$-connected. In this case, if $e_1 \sim e_2$, then $e_1$ is not on any cycles of $G - e_2$, which means it is a bridge; therefore $G - \{e_1, e_2\}$ is not connected. Therefore $e_1 \sim e_2$ if and only if $\{e_1, e_2\}$ is an edge cut of $G$.