Staggered fermions, Karsten-Wilczek (KW) fermions and Borici-Creutz (BC) fermions all retain a remnant chiral symmetry. The price to be payed is that they are doubled, and the resulting taste symmetry is broken by cut-off effects. We measure the size of the taste symmetry violation by determining the low-lying eigenvalues of these fermion operators in the two-dimensional Schwinger model which admits, like QCD, a global topological charge of a given gauge configuration. A first result is that it matters whether the pertinent eigenmode is a would-be zero-mode or a non-topological mode. The intra-pair splittings of the fermion formulations mentioned are found to depend sensitively on the gauge coupling $\beta$. Still, it turns out that it is surprisingly difficult to verify standard Symanzik scaling for these taste-breaking effects.