If $p$ is a permutation whose disjoint cycle decomposition is $c_1c_2\cdots c_r$ (including cycles of length $1$), then the multiplicity of $1$ as an eigenvalue of the map induced by $p$ is precisely $r$. It follows that $p$ is a reflection iff there is exactly one cycle of length $2$ and all the rest are of length $1$; of course, this happens exactly when $p$ is a transposition.
Of course, we have to check my initial claim on the multiplicity of $1$.
Consider the matrix $M$ of the map induced by $p$. It is easy to see that, up to permuting the basis, $M$ is in fact a diagonal block matrix, with one block corresponding to each of the cycles $c_i$. The multiplicity of $1$ as an eigenvalue is therefore the sum of the multiplicities of $1$ as eigenvalues of each of the permutation matrices corresponding to the $c_i$. We are thus reduced to showing that if $p$ is the cycle $(1,2,3\dots,n)$, then $1$ is a simple eigenvalue. But this is immediate: the characteristic polynomial, which coincides with the minimal one, is $x^n-1$.