Is it true that bicarbonate binds reversibly to the heme moiety of hemoglobin? Or do the heme portion only bind oxygen?

Bicarbonate does not bind to heme. It is a byproduct of hydrolysis of carbonic acid as a result of carbon dioxide binding to heme proteins (carbaminohemoglobin). In this process, an H+ ion binds with hemoglobin and bicarbonate (HCO3) is transported out of the RBC in exchange for a chloride (Cl-) ion.

Besides ions, oxygen (oxyhemoglobin), carbon dioxide (carbaminohemoglobin), and carbon monoxide (carboxyhemoglobin) are known to freely and reversibly bind with heme.