When (if ever) is it grammatically correct to start a sentence with "however"?

In my English essay, I used "however" to start the 2 different sentences below: "However backwards it may seem, the accused were often better off if they falsely pled guilty..." and "However, further scrutiny will reveal the obvious differences..." My paper was checked... show more In my English essay, I used "however" to start the 2 different sentences below:

"However backwards it may seem, the accused were often better off if they falsely pled guilty..."

and

"However, further scrutiny will reveal the obvious differences..."

My paper was checked by another student, who wrote that they were "fragments" and to start a sentence this way is incorrect. I disagree. What's right?
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