CoinSwitch can ask you to verify your identity at any time, see their terms of service:
We are committed to protecting our Users from fraud, scam, and other such financial crimes on the Platform. In order to prevent such activities, it is necessary to take measures to verify the identity of the customer and the authenticity of the transactions. You agree to share personal information including, but not limited to, your Name, Email Address, Contact Information, Telephone Number, Username, Government Issued ID, Date of Birth, Photographs, Source and Ownership of Funds and other information as per the specific requirements of CoinSwitch and as may be specified to you from time to time.
Look at this list for some alternatives:
https://bitcointalk.org/index.php?topic=5218464Godex.io looks like a good option but I would suggest you do your own research, and read the tos before anything.