こそすれ is a contrastive conjunction that pairs a positive statement with a negative one of the opposite nature. It is like saying 'though it may do X, it never does Y' or 'it does X, but not Y at all'. The pattern emphasizes that the second clause is absolutely not the case, often to correct a misconception. For example, '彼は謝りこそすれ、怒りはしない' means 'He might apologize, but he never gets angry' — the speaker stresses that apology is possible but anger is out of the question. The word こそ adds emphasis, and すれ comes from the verb する (to do) in its hypothetical form, but the whole pattern is a fixed expression. This grammar is quite formal and appears mostly in written language, such as essays, news articles, or speeches. In conversation, you'd more likely hear 'けど' or 'が' instead. It can also be used with adjectives and nouns, though verb usage is most common.