If you're a nerd like me, you can go through Hinds' Precedents and find 1800s vintage cases where judges and cabinet secretaries (albeit not presidents, afaik) facing impeachment were allowed to appear at House hearings and sometimes question witnesses. govinfo.gov/content/pkg/GP… 
However--and this is the crucial point--you can also look through Hinds Precedents and find lots of cases where the House conducted its investigation of similar officials entirely ex parte, without any involvement by the accused.
Both ways of doing impeachment investigations are valid. Why? Because the Constitution says the House gets to determine the rules of its proceedings.
The House of Representatives has discretion about how to conduct its business, including an impeachment investigation, guaranteed in the Constitution, and White House Counsels and Presidents have no power to tell it what to do or declare its procedures illegitimate.
