if you reported the exchange correctly, the starting basis of the replacement property is reflected on the 8824 and is what you should have been using for depreciation purposes since the exchange (of course less the allocation to land). To that would be added subsequent additions and improvements that were capitalized. so you already have the original basis which is what you are seeking. we don't know what was included in what you refer to as relinquished property equity but It is probably not the amount you should have used for basis.
"Is this what the IRS thinks was the basis at the time of the exchange?" I would say not. This is not standard terminology and is reflected nowhere on the 8824. Search the web and see if you can find a definition. I could not. Maybe ask a lawyer.
