As you found it is very important to sum the two stereo channels in your air chain when transmitting in mono. This should be done following your on air player and just ahead of the mono Orban processor if you want to maintain stereo files in your system rather than converting the files to mono, which is what I seem to hear you saying here.
As far as inverting the phase of one channel, this should only be done if needed to correct a phase problem. A CD mastered and marketed with this problem is very rare and would be considered very defective.
From: Robert Chrysafis [mailto:email@example.com]
Sent: Thursday, April 13, 2017 6:45 AM
To: The BROADCAST eList; CRTech
Subject: [CRTech] phasing issues - Fixed
Think i have found the workaround for the out of phase audio issue.
flipping one of the tracks in audacity is not an available option in that particular program but rendering stereo tracks to mono is.
it seems rendering the problem tracks to mono brings all the missing vocals and instrumentation back.
i have experimented with this option on two or three songs and it appears to work.
not the solution i would have liked but it works and still sounds good.