Clement’s theorem on twin primes

Theorem. (P. Clement) Given a prime number $p$, $p+2$ is also a prime (and $p$ and $p+2$ form a twin prime) if and only if $4(p-1)!\equiv -4-p(mod{p}^2+2p)$.

Richard Crandall and Carl Pomerance see this theorem as “a way to connect the notion of twin-prime pairs with the Wilson-Lagrange theorem.”