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I use powder Tide laundry detergent. Its cheap and works great! It provides plenty of grit to scrub grease. Works best with warm water as cold will leave a bit of a slimy film until it drys.
I'll bet Tide is great for your skin. I need this stuff throughout the day. The stuff we used at work was excellent but I guess they are trying to cut cost and bought the cheapass gojo that you can buy at Advance which is crap.
During the day, in the shop I use Simple Green Hand Cleaner. At the end of the day I use Dawn dishwashing liquid. I've found that if I use solvent based cleaners (Go-Jo, Permatex, etc.) all day long my hands crack and bleed.
I start with Gojo, then while hands are still covered in it (and before adding any water) go to good old Boraxo powdered hand soap.
Only when the Boraxo is starting to cake and fall off do I add a tiny bit of water and scrub more. Then I add more water and finish cleaning. If my hands are still not clean, then a bit more Boraxo & water usually do the trick.
Of course the best way to clean your hands is to keep them moisturized in the first place. I never remember to do this unfortunately, so my dry hands tend to soak up whatever grease and oils I come in contact with and all the dirt stays in the cracks and wrinkles for days.
Purchased a box of the rubber gloves and found it has save me considerable time cleaning, not only the hands, but also under the fingernails, and although they are thin, they have saved a knuckle on occasion. I just need to remember to put them on before starting a project. I look for the thickest type made, don't like the kitchen type rubber maide found them too restrictive. Regarding a hand cleaner, if there is any Pors 15 'marine clean' setting around I will use that, also good for cleaning grungy parts.
Purchased a box of the rubber gloves and found it has save me considerable time cleaning, not only the hands, but also under the fingernails, and although they are thin, they have saved a knuckle on occasion.... Eastwood Co. - Nitrile Gloves Disposable Medium
FYI: Nitrile gloves like the ones you list are fine, but be aware that you shouldn't use latex gloves for working on cars & trucks where you're coming into contact with fuel and other solvents. It breaks down the latex quickly and if done regularly you can develop an allergy to latex.