Can I invoke a template for print media?
You cannot tell in drupal, if your page is going to be printed.

CSS and media queries is browser based. You deliver the same content (html) and the browser will generate different output for display and print.

If you want to do deliver a special print version, you have to provide a different url for this and link it on your page, for example a print button. You can then check in the twig for the special url and change the html.