Directed Graphs and Network Flows · Challenge problem

A tournament on teams $1,\dots,5$ has team $i$ beat team $j$ exactly when $j-i \equiv 1$ or $2 \pmod 5$. A Hamiltonian cycle visits every team exactly once and returns to its start, always moving winner-to-loser. Counting two cycles as the same only when they visit the teams in exactly the same cyclic order and direction, how many Hamiltonian cycles does this tournament have?