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Yes, after the expiry of your H-1B visa, you can work for a US company remotely from India, as a freelancer or an employee. However, since the work is performed here and you receive your income in India, it will be taxable here.
Generally, you do not need to pay tax in both India and the USA; it is only taxable in India. Further, to avoid tax withholding in the USA, fill out Form W8BEN or W8BEN-E. It serves as evidence that your income is not taxable in the US. Additionally, if any of your US tax is withheld, under the foreign tax credit, you can claim that amount during your ITR filing.
The salary from a US company is taxed in India as Indian-sourced income and as per Indian income tax slabs. It is because the work is performed in India, and the income is also received here.
In case your US employer has deducted tax, or you have earned income in the US, then you may need to file a US non-resident tax return (Form 1040-NR) to report the income or claim a tax refund on it.
To avoid double taxation on your income, under the DTAA signed between the USA and India, you can claim a foreign tax credit in India. For this, you need to fill out Form 67 and submit proof of your paid US taxes at the time of filing the tax return.