  Gauge theory compactified on an orbifold is defined by gauge symmetry, matter
contents, and boundary conditions. There are equivalence classes (ECs), each of
which consists of physically equivalent boundary conditions. We propose the
powerful necessary conditions, trace conservation laws (TCLs), which achieve a
sufficient classification of ECs in U(N) and SU(N) gauge theories on $T^2/Z_m$
orbifolds $(m=2,3,4,6)$. The TCLs yield the equivalent relations between the
diagonal boundary conditions without relying on an explicit form of gauge
transformations. The TCLs also show the existence of off-diagonal ECs, which
consist only of off-diagonal matrices, on $T^2/Z_4$ and $T^2/Z_6$. After the
sufficient classification, the exact numbers of ECs are obtained.
