conjunction
Conjunction used to introduce a statement that contrasts with what one would expect from the preceding context. Often carries a tone of mild criticism or surprise.
彼は何でも知っていると言う。そのくせ、簡単な質問にも答えられない。
He says he knows everything. And yet, he can't even answer a simple question.
彼女はダイエット中だと言
She says she's on a diet. Even so, she eats cake every day.
それなのに is similar but more directly expresses 'despite that' and is slightly more formal. そのくせ often implies a critical or ironic tone.
にもかかわらず is a more formal and written expression meaning 'in spite of'. そのくせ is conversational and often carries a nuance of personal judgment.
Compound of その (that) and 癖 (habit, tendency). Literally 'that habit', it developed into a conjunction meaning 'and yet' by implying a contradictory habitual behavior.