Point sequences reaching 4-4 under deuce rules

In a game between two players, each point raises the winner's score by one, starting from $0$-$0$. Reaching $4$ points first wins the game, except that once the score has been $3$-$3$ the winner must lead by two. How many different point-by-point sequences lead to the score $4$-$4$?

$40=2\binom63$. A $4$-$4$ score is only possible if the game passed through $3$-$3$ (otherwise someone would have reached $4$ with a lead of two or more and won). There are $\binom63=20$ orderings of the first six points, then the next two points must be split one each, in $2$ orders.