  A systematic program is developed for analyzing and cancelling local
anomalies on networks of intersecting orbifold planes in the context of
M-theory. Through a delicate balance of factors, it is discovered that local
anomaly matching on the lower-dimensional intersection of two orbifold planes
may require twisted matter on those planes which do not conventionally support
an anomaly (such as odd-dimensional planes). In this way, gravitational
anomalies can, in principle, tell us about (twisted) gauge groups on subspaces
which are not necessarily ten-, six- or two-dimensional. An example is worked
out for the case of an $S^1/{\bf Z}_2\times T^4/{\bf Z}_2$ orbifold and
possible implications for four-dimensional physics are speculated on.
