Washing up liquid is a degreaser, normal handwash isn't. If you are getting dermatitis from the washing up liquid, you probably shouldn't be getting your hands covered in oil and grease in the first place
That doesn't really make sense, as a contact dermatitis from WUPL would be caused by it degreasing the skin. If you're using sugar etc to get really clean then any detergent (like a handwash) would probably do to help breakdown the inevitable dirt on the skin as well as any grease & oil.
As it is, I don't get contact dermatitis from WUPL (womens work, I don't do that) but have had it from the anti-bacterial handwashes at work (nurse, so these are not the same as the anti-bact stuff in the supermarkets, they are stupidly harsh).
Agree with the use of cheapie gloves when mechanicing on a bike, prevention better than cure and all that etc.