We have studied the magnetic properties of multilayers composed of ferromagnetic metal Co and heavy metals with strong spin orbit coupling (Pt and Ir). Multilayers with symmetric (ABA stacking) and asymmetric (ABC stacking) structures are grown to study the effect of broken structural inversion symmetry. We compare the perpendicular magnetic anisotropy (PMA) energy of symmetric Pt/Co/Pt, Ir/Co/Ir multilayers and asymmetric Pt/Co/Ir, Ir/Co/Pt multilayers. First, the interface contribution to the PMA is studied using the Co layer thickness dependence of the effective PMA energy. Comparison of the interfacial PMA between the Ir/Co/Pt, Pt/Co/Ir asymmetric structures and Pt/Co/Pt, Ir/Co/Ir symmetric structures indicate that the broken structural inversion symmetry induced PMA is small compared to the overall interfacial PMA. Second, we find the magnetic anisotropy field is significantly increased in multilayers when the ferromagnetic layers are antiferromagnetically coupled via interlayer exchange coupling (IEC). Macrospin model calculations can qualitatively account for the relation between the anisotropy field and the IEC. Among the structures studied, the IEC is the largest for the asymmetric Ir/Co/Pt multilayers: the exchange coupling field exceeds 3 T and consequently, the anisotropy field approaches 10 T. Third, comparing the asymmetric Ir/Co/Pt and Pt/Co/Ir structures, we find the IEC and, to some extent, the interface PMA are stronger for the former than the latter. X-ray magnetic circular dichroism (XMCD) studies suggest that the proximity-induced magnetization in Pt is larger for the Ir/Co/Pt multilayers than the inverted structure (Pt/Co/Ir), which may partly account for the difference in the magnetic properties. These results show the intricate relation between PMA, IEC, and the proximity-induced magnetization that can be exploited to design artificial structures with unique magnetic characteristics.