The sign is multiplicative.

Theorem

 For all g, h Sn, we have sgn(gh) = sgn(g)sgn(h).

We also say that sgn is a multiplicative map from Sn to {1, -1}. (The notion morphism explores this view further in a general context.)
The theorem implies the following way of determining the sign.

Corollary

 If a permutation g is written as a product of cycles, then sgn(g) = (-1)w, where w is the number of cycles of even length.