All greyhound owners are aware of the awesome “roaching” their hounds do. It is something that is almost endemic to the breed.
Of course, it has an origin. When cockroaches die (from poisoning or natural causes), they end up on their backs with their feet in the air.
Greyhounds, when they are on the track, sleep in a small-ish crate, and to stretch out, they universally learn to sleep on their backs, with their legs in the air.
The comparison to the dead cockroach is inescapable, hence, the “roach”