Can the review of a tenure track application start before the reference letters arrive?
At least in my department, yes, faculty candidates can be interviewed before their reference letters arrive, although an actual job offer requires all letters.  (In fact, this is relatively common for more senior applicants, who often want to keep their application confidential as long as possible.)  But that was a policy decision by the department, made with the explicit approval of higher administration.  Other departments and other universities can set their own hiring policies/procedures/requirements.
The overarching rule (according to my understanding of US equal-opportunity labor laws) is that the hiring process must be consistent; the same process must be followed for all applicants.  If you interview one applicant before all their letters arrive, you cannot then claim that you didn't interview a second applicant because you didn't have all their letters.
(I ran my department's faculty hiring committee for three years.)