Vinyl distortion - What's the problem?

The right channel of my turntable output is a little distorted, especially in certain upper-mid frequencies, while the left is crystal clear. I have a shure v15 (whatever the brand-new version is designated). I've tried replacing the stylus to no effect and the problem switches sides when I switch the left and right input cables around, so I know it's not my amp or phono input. It does this on brand new and well worn records. Any suggestions?
Sounds like you need to check your cartridge alignment, and your anti-skating setting, and perhaps the azimuth. We don't know what TT or tonearm you have, so it's hard to speculate any further than that.
As TWL says, go over all the basics of the cartridge mounting process. I would add go slow, don't assume it was done right the first time. Make sure your tt is properly levelled. Are you using the brush/stabilizer thing (whatever it is called) on the Shure? You have to adjust the tracking force accordingly. In tracking force, a tiny bit more is better than a tiny bit less. I can't remember if the anti-skating has to be set exactly for the higher setting of the tracking force if you are using the stabilzer/brush thing, check the instructions provided by Shure. Good day.
Sorry I forgot to say the first time that the cartridge is on a brand-new Rega P25 with the stock 600 tonearm. I checked the alignment with the shure alignment card that came with it and set the tracking force for the recommended 1.5 grams with the brush/stabilizer. I don't know what azimuth adjustment is, so I suppose I didn't adjust that. I'll check with a level tonight to see if it is sloping any direction. Looks fine to the eye, but maybe that's not good enough.
One other thing you might check, is to see if your stylus cantilever is bent. You can check this by looking at it directly from the front when it is on the arm rest. It it does not look like it is centered, then it could be bent. This would require a new stylus assembly.

You don't have to worry about adjusting azimuth on the Rega. Make sure the anti-skate is set to the same setting as the tracking force.
try tuning out the distortion by adjusting the anti-skating force +/- the optimum setting designated by the cartridge mfg. the exact setting proscribed isn't always the best on any given arm/cartridge combo.
The procedures suggested above will probably identify your culprit, but if not, you should also try to rule out any possible problem with the electronics/cables chain the TT is feeding (I'll assume you have already ascertained that the problem is not present on any other source besides phono).

You can do this be reversing the leads coming from the TT at the input to the phono preamp. If you reverse the leads, and the problem remains in the same channel, then your distortion is originating somewhere between your phono inputs and your preamp outputs.

However, if you try this and the problem switches channels, but nothing suggested above has helped, then suspect something in the lead-out cables or jacks themselves, with the jacks being the most likely.
As you've already tried switching channels, as Zaike suggests, it's probably either: a) the electrical connection in the cartridge-to-phono chain (have you checked the pins on the cartridge?) b) a problem with the cartridge. So, Twl has a point -- if not, then it could be a problem with the coils. (I hope not). Good luck!
You caught me, Greg, thanx. Sorry, it's too late at night! My last paragraph still stands, however.
All the previous posts had good advice. The only thing I would add is to check the cable. Is it worn or in need of replacement? Is it attached securely?
Thanks for all the input. I adjusted the "bias" slider (is this the same thing as anti-skating?) as suggested by Newbee to be around .75-1 units instead of the 1.5 to match the tracking force as recommended by Rega. Seems to have helped a lot. Occasionally, there still seems to be a little distortion as the arm gets onto the inner part of the record, but I'll try to tweak my alignment to see if I can eliminate that.
sounds like you have good ears and are paying attention to what you're doing. you may now be hearing distortion in the record itself - i have a lot of records that distort near the end. i believe this occurs in the cutting process and there is nothing you can do to correct it. if its a set up problem you should hear this on most disc's (and i'll bet you don't). If possible i try to stay away from records which are over long on one side for just that reason.