I also aggree running with the grain works better. Just think of the grain as tiny little slots/grooves. If the piece is spinning, you are only hitting the tops, so it does'nt get down in there, and pull as much dirt up, but going in the same direction as the grain gets down in there alittle better, and pulls dirt out better.
I have used orange cleaners also, well years ago anyway, and they worked well, though I don't really use them now. I have also used butcher block cleaner. Don't know If I have ever heard of anyone using that on shafts, but it cleans alright also, however I do not use it anymore either. The issue with me is- the amount of water it takes to clean these products out of the shaft does'nt sit well with Me, but that's just personal preference on My part. I have never had a shaft warp from using them, but I was always carefull to use the least amount of water needed to get the stuff all out anyway, and even sometimes used a hair dryer to dry them well afterwards (not something I do anymore, but never had a shaft warp from It back when I did).
If using a product that foams, and lifts the dirt that way, the magic erasers are also good for that, because By sqeezing the pads, they foam up well, and makes It easier to keep mostly foam on the shaft and not pure water, or solvet. I would be very carefull using the pads wet while spinning in the lathe though, because I have had them grab & stick to the shaft before, and if it had a hold of a finger when that happened, might not be a good thing
There are many ways to clean a shaft, some shafts require more drastic measures then others to clean, and some of the ones I've done were basically restorations, so for me It just depends on what It takes to get any given shaft clean, as to what product i use on it, and how I go about it. It seems like each shaft can almost have a life of It's own sometimes, as far as having different grain characteristics go & what type of grime you need to get out when it comes to cleaning. They might all be the same type wood, but they all sure don't clean up the same from My experiences.
I have a customer that owns a place with a kitchen in It, he would sit his cue in there in between matches while he worked, so the thing was covered in grease. In fact I even had to re-attatch his forearm to the handle of his cue, because it seems the grease soaked in and loosened the glue, then the forearm screwed right off the handle, and the glue was melted And sticky. what a mess that was to clean off.
Honestly what I did was just try different products out on My personal shafts at first, and picked up different methods of cleaning that way. There's no telling what's out there now that none of us have even tried on shafts yet.
Greg