  We study instanton partition functions for N=2 superconformal Sp(1) and SO(4)
gauge theories. We find that they agree with the corresponding U(2) instanton
partitions functions only after a non-trivial mapping of the microscopic gauge
couplings, since the instanton counting involves different renormalization
schemes. Geometrically, this mapping relates the Gaiotto curves of the
different realizations as double coverings. We then formulate an AGT-type
correspondence between Sp(1)/SO(4) instanton partition functions and chiral
blocks with an underlying W(2,2)-algebra symmetry. This form of the
correspondence eliminates the need to divide out extra U(1) factors. Finally,
to check this correspondence for linear quivers, we compute expressions for the
Sp(1)-SO(4) half-bifundamental.
