Bipartite perfect colorings of the infinite square grid: twin colors and the covering case

A perfect coloring ( equitable partition) of a graph $G$ is a function $f$ from the set of vertices onto some finite set (of colors) such that every node of color $i$ has exactly $S(i,j)$ neighbors of color $j$, where $S(i,j)$ are constants, forming the matrix $S$ called quotient. If $S$ is an adjacent matrix of some simple graph $T$ on the set of colors, then $f$ is called a covering of the target graph $T$ by the cover graph $G$. We characterize all coverings by the infinite square grid when the target graph is bipartite, proving that every such coloring is either orbit (that is, corresponds to the orbit partition under the action of some group of graph automorphisms) or has twin colors (that is, two colors whose identifying keeps the perfectness of the coloring). The case of twin colors is separately classified.