Diario de stevejones

If you find a cholla that just looks weird compared to its neighbors chances are it's an intergrade - either a two-parent hybrid or some level of back-crossing to one parent or the other*. One helpful aid to finding the relationship is to ID the taxon of the chollas that are found nearby. When encountering the odd ones make a note, and if possible separate observations, of the cholla taxa in the area.
There are numerous records of cholla intermediates in herbarium collections; many of those are noted by Donald Pinkava in this Canotia article.
*Or to a third taxon - theoretically possible

Lycium exsertum is flowering again locally, giving an opportunity to observe pin and thrum heteromorphy. The same condition can also be seen in L. fremontii flowers.

@gitanomad posted a comment on his observation here of Cylindropuntia fulgida var. mamillata curious as to why var. mamillata has been observed worldwide (see map), odd for a variety of a New World species. Spot-checking the observations from Africa and Australia, they all differ quite a bit from what is locally identified as that variety (key to varieties here). The stems are thicker, the tubercles denser and the stems often fasciate, sometimes cresting (a form of fasciation).
I don't know its provenance, but it appears someone found a fasciating var. mamillata, took joints or cuttings and started cultivating it, marketing it as "boxing glove cactus".
Cuttings made their way to Australia, southern Africa and Iberia where they escaped cultivation and became a bit of a pest. There are articles online on biological control using cochineal insects.
There are some variations in the plants found abroad; some seem to have given up fasciation but still carry the characters above - thick stems and dense tubercles.

One of my favorite plant common names is wingnut cryptantha, Cryptantha pterocarya. Arizona Flora treated them as two varieties separated by the number of winged nutlets - four winged nutlets => var. cycloptera vs three winged and one not winged => var. pterocarya. They have since been treated as species and Jepson has a key - see couplet 29. Without nutlets it appears plant color is a good hint - yellow-green for C. cycloptera, gray-green for C. pterocarya. In my experience the calyx lobes are a good spot to check for color. Further, pedicel length can help, longer in C. cycloptera.