Can H1 start company in India

[nina_summa]

I'm on an H1 with company A.
I have a registered company in India and plan to develop that.
The work will involve projects here as well as in India.

1. Can I work for my company in India simultaneously while on H-1 here in US?

The other option is that I use another name (not registered, just a website, business cards) as a subsidiary (or similar) of Company A. Are there legal issues with that?

I want to build up my client base and then move to my company in India.

 

[nina_summa]

seeking some more clarifications.

I have an H-1 with company A.
What if theres no income to the company in India (on paper or otherwise). What if I waited till my GC, would it still not be feasible?

So effectively, I would need an H-1/work authorization even to promote my family busines(in India) in the US or elsewhere, is that right? and cannot work on projects that could get paid by US customers.

Thanks.