Trace of Hermitian involutionary matrix

Question

let $V$ be a Hermitian matrix s.t. $V^2 = I$. Because it's Hermitian, we know that the trace is real. But as it is not necessarily PSD, what does it being involutionary tell us about the trace? I am unsure about the trace of the inverse: $$tr(V) = \sum_i \lambda_ i = tr(V^{-1}) =\sum_i \lambda_ i^{⁻1} \Longrightarrow |\lambda_i |= 1?$$