  In five- and six-dimensional $U(N)$ and $SU(N)$ gauge theories compactified
on $S^1/Z_2$ and $T^2/Z_3$ orbifolds, we propose a new method to classify the
equivalence classes (ECs) of boundary conditions (BCs) wihtout depending on the
structure of gauge transformations. Some of the BCs are connected through gauge
transformations and constitute ECs, each of which contains physically
equivalent BCs. Previous methods for classifying ECs have been used specific
gauge transformations. In this paper, we show that a geometric property of
orbifolds significantly narrows down the possibilities of connecting BCs and
completes the classification of ECs.
