It's generally a good idea to check them yourself as well. HMRC does sometimes make mistakes with people's tax codes, resulting in the wrong amount being taken off your pay, and your employer doesn't (and can't) correct those mistakes for you. Usually when there's a mistake it will eventually be corrected automatically by HMRC but that can leave you with an unexpected extra tax bill (or a refund, which is usually less of a problem), but those corrections happen months after the end of the tax year.

Also if you pass a certain income threshold you're required to fill in your own tax return anyway.