As far as starting a company on H1-B is concerned, YES!, you can start your own company.
Starting a business in US is surprisingly simple — there are various legal structures under which you can ‘start business’. A sole proprietorship, a partnership, a corporation and a limited liability company are few of the legal structures under which an individual can choose to conduct a business. However, each of these entities is entitled for different purposes, a subchapter ‘C’ corporation is the entity that best suits you from a legal standpoint.
A corporation is owned by shareholders. Corporations must have at least one owner, the owners are called shareholders or stockholders. There is no restriction for an individual on a H1-B status from owning a stock in a corporation and also, no upper limit on how many stocks a person is allowed to own. That being said, individuals on H1-B status can rest easy by taking a passive role in owning a company – Passive meaning that they are not directly involved in day-to-day operations of the company. Being a shareholder or a board member is certainly a passive role. They do not directly involved in running a business. From a legal standpoint, I do not see a reason why you can not start your own company – Planning for the company, making owner decisions, and hiring employees who acutually take care of the day-to-day operations is totally acceptable.
So now comes the tricky part, you are on H1-B status and decide to work for the company you just started then the company will have to sponsor your H1B visa. However, USCIS wants to be sure that H1B petition filing company ensures that the employer-employee relationship exists between the petitioner and beneficiary and the company possess the right to control the beneficiary’s work including ability to hire, fire and supervise. The USCIS also states that an employee who owns a majority of a sponsoring company and reports only to him/herself may not be able to establish a valid employment relationship for H-1B purposes.
That being said, based on the visa status, you can start a company in the US. However, using that company to sponsor your H1-B is nearly impossible because of the fact that the arrangement does not ensure proper employer/employee relationship. Evidently, as a company shareholder you will be considered as an owner entitled to receive profits/dividends but receiving any compensation as a regular employee is strictly not allowed. The new company you started could sponsor a H1b visa but not yours.