Why can't applicants ask for letters of recommendation on potential employers/bosses?
Short answer: supply and demand. Established universities are comfortable that they can get qualified applicants even with requiring letters; candidates would have a very difficult time finding a job if they required their employers to submit letters of recommendation to them. 

The early stage (making a short list) is asymmetrical for sure, but it's sort of apples and orange.


There are only so many departments that work in a particular subfield, so the applicant is likely already familiar with the department and potentially even with the faculty members in that department. Thus, letters wouldn't provide much new information. (Possible exception for small R&amp;D companies rather than universities, but such jobs rarely require letters). 
Conversely, there are many applicants for any particular slot, and the hirers will know very few of them. Thus, without letters, there are few practical, effective ways to know which candidates should interview. 


In later phases, the process does become more symmetrical. Both sides tend to make their own judgments at this phase rather than relying on reputation or letters. 


These interviews should be both ways -- I would be surprised if any hirer would hesitate to discuss their interests, backgrounds, expectations, and style with a candidate they are seriously considering hiring.
You are also likely to have a chance to meet with the supervisor's group, colleagues, and employees at some point. If not, no reason you couldn't request such a meeting, or just set one up directly.