We demonstrate that high anisotropy of elastic constants of chromonic liquid crystals leads to a number of spontaneously twisted nematic director fields around colloidal particles in these non-chiral fluids. For spherical colloidal particles with surface inducing degenerate planar nematic ordering we observe that boojum defects at the particles' poles acquire twisted internal structure, extending up to three particle diameters along the rubbing direction of the cell. The twist handedness of the two boojum defects at the poles of the particle can be either the same or opposite, and we can switch the defects handedness by localized thermal microquenching. Numerical simulations confirm that the transitions into the distorted states are induced by lowering of the twist elastic constant, which results in two (meta)stable chiral configurations of the boojums, separated by an energy barrier much higher than the thermal energy. We show that boojum handedness can change the pairwise elastic interaction between the two particles positioned along the rubbing direction from repulsive to attractive.