Is it possible to be Multi-Partial, not Impartial in Mediation?

     Sometimes I feel like I am being torn in two.  My "empathy" quotient goes up so high that I truly feel like I am on each side of the dispute I am mediating for the time I am in each room.  Then when I go into the other room, the other side wins me over completely.  Is this the kind of neutrality that is effective to settle disputes?  How do others keep themselves from being multi-partial, or is it actually a good quality to have when the facts or issues call for it?
     Last week, I mediated a dispute arising out of an allegation of disability discrimination by a long-term employee against a small business.  I empathized with the Plaintiff, because she had loved the job and got no full or legitimate explanation for why she was terminated, leaving her lawyer to conclude the stated reason of insubordination was pretextual and the real reason was her recent diagnosis of an old injury, which now required surgery.  In the other room, I truly empathized with the employer, whose office manager had made a quick decision in reaction to an argument with the employee and who now was looking at paying many multiples of the Plaintiff's salary to avoid a trial on this case.
     I am confident that at the end of the day, both clients and both lawyers saw me as their champion--and the key to a successful negotiation.  I was able to confidently reassure them that the result was fair in the scope of the litigation before them.  But is that what neutrality is supposed to look like? Is a mediator supposed to remain neutral and impartial or is it actually more effective to be "multi-partial"?