I have been working in the U.S. on an H-1B for one year now. I want to create an Internet start-up company with my cousin back home in India. All my work would be online; I wouldn't need to move back to India. None of our clients would be in the United States. I could do this part-time (mainly on weekends) and there would be no conflict with my current employer. I know I'm not supposed to work in the U.S. without specific authorization, but this wouldn't be the same, right? Would I get in trouble with immigration?
This is a delicate question. It is unclear whether you ought to get in trouble for the specific kind of work you have in mind (actively working for a company that is clearly not U.S.-based), but it is safe to assume that you could.
Remember: As an H-1B visa-holder, you are allowed to work in the U.S. only for employers who have filed an I-129, Petition for a Nonimmigrant Worker with U.S. Citizenship and Immigration Services (USCIS) on your behalf.
Of course, if you are working for yourself in India, you might argue that you are not even working for a U.S. employer. The problem is that the definition of "work" (or employment) "in the U.S." for purposes of U.S. immigration law is somewhat ambiguous: The term "employment" means "any service or labor performed by an employee for an employer within the United States," and the term "employer" means a person or entity "who engages the services or labor of an employee to be performed in the United States." (See the U.S. Code of Federal Regulations, at 8 C.F.R. § 274a.1.)
You might expect that the main concern of the U.S. government, when it comes to regulating the employment of foreigners, would be with limiting their access to U.S. employers—including any foreign company with a subsidiary, branch office, or agent in the United States. (Much of immigration law is, after all, directed at protecting opportunities for U.S. workers.)
However, U.S. immigration agencies have interpreted their powers broadly and, as a result, most immigration lawyers would probably tell you that "employment in the U.S." includes any work performed by any person present in the U.S., regardless of the person's citizenship and regardless of the location of the company (or the company's clients).
If "employment in the U.S." includes any work performed by any person present in the U.S., this means that you should not work for your Indian-based company without first either filing a new H-1B petition for concurrent part-time employment with the company, or abandoning your current H-1B employment and changing your nonimmigrant status to L-1 (Intracompany Transferee), or obtaining a green card. (The first option would be the most convenient in most regards; unfortunately, it might not be the easiest.)
If none of these three options seems achievable within a reasonable time, you would need either to choose between your current H-1B employment and your Indian company, or to prepare a very sophisticated legal defense strategy.
Under any scenario, you might want to secure the assistance of an experienced immigration attorney.
A final note: Don't forget that at tax time, you'll be required to report your entire income to the IRS, from sources both inside and outside the United States.