Let $t_a$ be the Dehn twist about a circle a on an orientable surface. It is well known that for each circle b and an integer n, $I(t{ⁿ}_a\left(b\right),b)=\left|n\right|I(a,b)²$, where I(·,·) is the geometric intersection number. We prove a similar formula for circles on nonorientable surfaces. As a corollary we prove some algebraic properties of twists on nonorientable surfaces. We also prove that if ℳ(N) is the mapping class group of a nonorientable surface N, then up to a finite number of exceptions, the centraliser of the subgroup of ℳ(N) generated...