Set up CI and pre-commit hooks for `clang-format`

Essentially the title. For example, the LLVM project has [a CI workflow](https://github.com/llvm/llvm-project/blob/main/.github/workflows/pr-code-format.yml) that checks if the PR is correctly formatted, and what fix to apply. I understand `git clang-format` is also a supported subcommand of git, so we could find some ways to prevent badly-formatted code from even being committed, let alone pushed.

Start by reviewing the LLVM example at .github/workflows/pr-code-format.yml and the repository's existing CI configuration. Investigate how git clang-format can support a pre-commit check, then define CI and hook behavior that identifies formatting problems and shows the applicable fix; done means both checks are set up and documented.

Written by the indexing model from the issue text.