I've found a way to "drag" oil and grime off the sidewalls with a stiff nylon brush. Brush must be applied sideways with your thumb pressing the bristles hard down on the tire. The action feels like I imagine waxing off hairs does. There's a lot of resistance, then snap the brush drags a whole flake of grime off. Doesn't abrade the tire at all.

With this I just got a pair of White Paselas sparkly clean using a detergent no stronger than dish soap. Wish I discovered this years ago.