  An occurrence of a consecutive permutation pattern $p$ in a permutation $\pi$
is a segment of consecutive letters of $\pi$ whose values appear in the same
order of size as the letters in $p$. The set of all permutations forms a poset
with respect to such pattern containment. We compute the M\"obius function of
intervals in this poset, providing what may be called a complete solution to
the problem. For most intervals our results give an immediate answer to the
question. In the remaining cases, we give a polynomial time algorithm to
compute the M\"obius function. In particular, we show that the M\"obius
function only takes the values -1, 0 and 1.
