The electromagnetic E1 and M1 transitions in heavy quarkonia ( c c̄,b b̄,c b̄) and the magnetic moment of the Bc± are calculated within the framework of the covariant Blankenbecler-Sugar (BSLT) equation. The aim of this paper is to study the effects of two-quark exchange current operators which involve the Q overlineQ interaction, that arise in the BSLT (or Schrödinger) reduction of the Bethe-Salpeter equation. These are found to be small for E1 dominated transitions such as ψ( nS)→ χcJγ and ϒ( nS)→ χbJγ, but significant for the M1 dominated ones. It is shown that a satisfactory description of the empirical data on E1 and M1 transitions in charmonium and bottomonium requires unapproximated treatment of the Dirac currents of the quarks. Finally, it is demonstrated that many of the transitions are sensitive to the form of the Q overlineQ wavefunctions, and thus require a realistic treatment of the large hyperfine splittings in the heavy quarkonium systems.