I misunderstood. Deleted.

You've described an approach when $r=n$. But I think the OP is also interested in $r<n$. (Although I'm not yet convinced that any reversing is necessary in the case you describe. I'll think about that a bit more.)

Maybe I'm misunderstanding the OP's question. I interpret it that there are 4 persons showing up for dinner but you only have 3 chairs to seat them at a round table. I think the question is to generate all possible seating arrangements (with just 3 being seated at a time).

That means one would first find all of the ways 3 out of 4 dinner guests could be chosen. For each of those sets of 3, all possible circular arrangements would be generated. For this example that would be (3-1)! = 2 ways for each set of 3.

So I see the question being general for $n \geq r \geq 2$. For $n=r \geq 2$, one just has to fix 1 dinner guest and then permute the rest as usual resulting in $(n-1)!$ arrangements. No need for reversing.

But the OP will need to clarify.

Lol. I'd love to be invited to your next party!

Per the question I was interested in r<n but then I want to believe that once it works for r<n it should work for r=n

I couldn't find any built in functions or options that would produce circle permutations. Here is a way to do it.

First, define a function that puts a permutation into some canonical order. I chose to rotate the permutation until its "minimal" element (by sort order) is first.

CanonicalizePermutation[perm_List] := 
  NestWhile[RotateLeft, perm, Not@*MatchQ[{1, ___}]@*Ordering]

If we take a list of permutations and put them all in this canonical form, we can then use Union or DeleteDuplicates or just DeleteDuplicatesBy.

DeleteDuplicatesBy[Permutations[{a, b, c, d}, {3}], CanonicalizePermutation]
(* {{a, b, c}, {a, b, d}, {a, c, b}, {a, c, d}, {a, d, b}, {a, d, c}, {b, c, d}, {b, d, c}} *)

DeleteDuplicates[CanonicalizePermutation /@ Permutations[{a, b, c, d}, {3}]]
(* {{a, b, c}, {a, b, d}, {a, c, b}, {a, c, d}, {a, d, b}, {a, d, c}, {b, c, d}, {b, d, c}} *)

Union[CanonicalizePermutation /@ Permutations[{a, b, c, d}, {3}]]
(* {{a, b, c}, {a, b, d}, {a, c, b}, {a, c, d}, {a, d, b}, {a, d, c}, {b, c, d}, {b, d, c}} *)