A root Tolkien used for “revolve” in notes from the late 1950s or early 1960s, along with variant √KWEL and derivatives Q. querend- “pivot, revolving center” and Q. querma “spinning wheel, turn-table” (PE17/65). It replaced √PEL for this purpose, which came to mean “edge, bound, fence, limit”. This note seems to imply that √KWER primarily meant a horizontal rotation, but it was also the basis for Q. querna, as in silmë nuquerna “s-reversed” for an inverted silmë tengwa (LotR/1123), though it may be that the nu- here is necessary to imply a vertical rotation.
The earliest precursor to this root was ᴱ√KERE “turn” from the Qenya Lexicon of the 1910s, though this root’s derivatives mostly had to with earthenware and pottery (QL/46). In the Early Quenya Grammar of the 1920s Tolkien introduced several variants of this root: ku̯ere, ki̯ere and elaboration ᴱ√kereke “turn round and round, send to and fro”, the latter said to be the basis of words having to do with “weave” in a syncopated form √kreke (PE14/65). In this document, none of these variants had any derivatives, but it seems the first of these ku̯ere survived in Tolkien’s later conception of the languages.
Neo-Eldarin: For purposes of Neo-Eldarin, I would mostly use √KWER for “revolve, ✱turn”, and avoid the variant √KWEL which (a) has no derivatives and (b) conflicts with √KWEL “fade”. However, √KWEL is useful for preserving Noldorin words having to do with rotation from the earlier sense of the root ᴹ√PEL “revolve on fixed point” from the 1930s, so I would keep it as Sindarin-only variant to allow the retention of words like N. pelthaes “pivot”; this may also have been Tolkien’s motive for having such a variant of √KWER “revolve”.