I would like to know if there's a special name for this kind of ordering. When I say there is a strict total order on $X/\sim$, what I mean is that two distinct elements in the same equivalence classes are considered equal, but an element of one equivalence class is either less than or greater than an element of another equivalence class.
In other words, what kind of ordering does the set of poker hands have? For example, one equivalence class is the set of straight flushes with a high card of 10 (i.e., 6 through 10 of the same suit). Any two from these four hands are considered equal. However, a hand from this set is greater than any poker hand that's less than a straight flush (e.g. four of a kind, straight flush, etc.) or a straight flush with a high card less than 10, and less than any straight flush with high card greater than 10.