Permutations Involving 8 Elements

Probability Level 5

How many permutations $\tau$ on 8 elements are there such that $\tau \circ \tau$ is the identity permutation?

Details and assumptions

You may think of a permutation on 8 elements as a way to shuffle a deck of 8 cards. The identity permutation would correspond to the shuffle where we do nothing (hence it stays the same = identity). If $\tau$ corresponds to only exchanging the top 2 cards, then $\tau \circ \tau$ would return the deck to it's unchanged state.