When we started Ximian, we immigrated Miguel on an O1 visa because he wasn't eligible for an H1-B. He had been in a couple of magazines (including Time in Latin America) and the magazine clippings were enough to qualify him for this visa. Unfortunately the O1 visa is a bit strange - it requires renewal every six months, and, we later discovered, is used mostly by musicians and dancers and other artists/performers.
So this is an unusual path but perhaps it will be useful to someone.
I am not a lawyer, but I have been through much of this recently. It gets complicated but there are apparently a number of options, so make sure to talk to a lawyer. Below is what I understand.
Starting a company in the US is possible on an H-1B. Just not 'working' for it. I am not clear on the exact definition of working, but it should be ok if you are not paying yourself. Just plan to have more equity in lieu of compensation.
Now figuring out how to pay yourself can be tricky. The main option that I know is to try to get an E-2 visa. This is what I have been recommended by my lawyer and ended up using. If you convert the money you put into the company along with your time (that you have already put in) this becomes a non-trivial investment. All you would need to figure out is show how the business can be bootstrapped, and now you are a successful investor wanting to work on the business that you have setup.
Alternatively, you can set up a child company 'back home' controlled by the parent company here, and have that child company employ and pay you.
Note that getting your startup applying for an H-1B visa for yourself might be impossible in the short term.
An H-1B is petitioned for by the company that is going to employ you. You can be a founder of that company, though.
Presumably you are going to quit the employer that is currently sponsoring your H-1B visa and get a new H-1B sponsored by the company you are founding?
Note that you will probably need to leave the US during this process (I did) and you will need to demonstrate that the new company isn't just a front to keep you in the country (helps if you have other founders, other employees, don't have a controlling interest, etc)
@pageman, have you seen http://www.startupvisa.com yet?
Reg. h1b: I believe if you can prove the company can pay your prevailing wage (either you've raised money, or the company is generating enough revenue) your company can indeed sponsor your H1b. You should check with your immigration lawyer.