  It is well-known that cross caps on surfaces in the Euclidean 3-space can be
expressed in Bruce-West's normal form, which is a special local coordinate
system centered at the singular point. In this paper, we show a certain kind of
uniqueness of such a coordinate system. In particular, the functions associated
with this coordinate system produce new invariants on cross cap singular
points. Using them, we classify the possible symmetries on cross caps.
