Nematic phase transitions in high-temperature superconductors have a strong impact on the electronic properties of these systems. BaFe2As2, with an established nematic transition around 137 K induced by magnetic fluctuations, and BaNi2As2, a nonmagnetic analog of BaFe2As2 with a structural transition in the same temperature range, share a common tetragonal aristotype crystal structure with space-group type I4/mmm. In contrast to BaFe2As2 where collinear stripe magnetic order is found for the orthorhombic low-T phase, a unidirectional charge density wave together with (distorted) zigzag chains are observed for the triclinic low-T phase of BaNi2As2. Here we show that between the high- and low-T phases of Ba(Ni,Co)2(As,P)2 an additional phase with broken fourfold symmetry and dxz orbital order exists which is a promising candidate for charge/orbital-fluctuation-induced nematicity. Moreover, our data suggest that the enhanced Tc found for Ba(Ni,Co)2(As,P)2 for higher Co or P substitution levels might result from suppression of the (distorted) zigzag chains by reducing the contribution of the dxy orbitals.