Firstly, cats arch their backs when they are fearful – it’s a pretty normal response for them, so they could be getting scared when you’re stroking them.
But..! Some cats arch their back because they enjoy the sensation of being stroked and they then try and push against your hand.
If your cat is purring deeply and it is also lifting it’s tail (and more importantly, not running away) chances are it’s the latter – it’s being aroused and is enjoying the feeling!
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