  We show that there is a bijection between real-linear automorphisms of the
multicomplex numbers of order $n$ and signed permutations of length $2^{n-1}$.
This allows us to deduce a number of results on the multicomplex numbers,
including a formula for the number of involutions on multicomplex spaces which
generalizes a recent result on the bicomplex numbers and contrasts drastically
with the quaternion case. We also generalize this formula to $r$-involutions
and obtain a formula for the number of involutions preserving elementary
imaginary units. The proofs rely on new elementary results pertaining to
multicomplex numbers that are surprisingly unknown in the literature, including
a count and a representation theorem for numbers squaring to $\pm 1$.
