Okay, since I was chatting with folks about it and we were working on house projects all day, I went ahead and washed my hair today! This is the first time in 6 weeks (ish). See this post for context.
I used about a quarter cup of baking soda mixed with a quarter cup of water for the cleaning of the hair. The baking soda, like on your stove, gets rid of grease, and after using it, did indeed leave my hair with that squeaky, lots of friction feeling that you get if you've ever used bar soap to wash your hair. The process itself I think I'll have to get used to. It doesn't foam or lather, and the mixture has low viscosity because of the water. I poured it over my head in bits, trying to mostly keep it on my scalp and get at the different parts. Then there was a bunch of baking soda not in solution that I just plopped on.
I think the water is mostly to be a conveyance rather than needing to dilute the baking soda at all. It's a buffer, so I'm not really worried about concentration, and you're in the shower so there's going to be more water everywhere. I think next time I might just have the powder with him and scoop it on mushy rather than with too much water. (I didn't actually go back and look up what I was supposed to do...)
After the baking soda was rinsed out I put on a quarter cup of apple cider vinegar mixed with a quarter cup of water. The reason for this I've read is that it neutralize the baking soda to keep you scalp in pH balance and also kills bacteria and stuff growing on your scalp, since it's an acid. Also it's supposed to be a natural natural something because it "removes build up and seals your hair cuticle." I'm not sure of the science, but it did work, making my hair feel smooth and clean again as if I'd used a conditioner. So all in all very good results. I used apple cider vinegar because I read somewhere that it's a little less acidic than white vinegar, and also because it doesn't smell as strong. I used what I had in my kitchen, so nothing special. It doesn't smell like vinegar to me at all, though my husband who can smell my head says it's faintly there. I certainly won't be knocking anyone over with it.
All in all, I'm pleased with the results. It's like I washed my hair or something. The real test will be how often I need it. The goal is less frequently than I had been needing to wash my hair with shampoo - ideally once a week or once every two weeks. That was the whole point of not washing my hair for six weeks, to make it so my scalp doesn't instantly produce grease as soon as it's cleaned. Keeping the pH balance to neutral with the base then acid rinse is supposed to help with that.