We conducted an extensive Monte Carlo study of factor-rotation local solutions (LS) in multidimensional, two-parameter logistic (M2PL) item response models. In this study, we simulated more than 19,200 data sets that were drawn from 96 model conditions and performed more than 7.6 million rotations to examine the influence of (a) slope parameter sizes, (b) number of indicators per factor (trait), (c) probabilities of cross-loadings, (d) factor correlation sizes, (e) model approximation error, and (f) sample sizes on the local solution rates of the oblimin and (oblique) geomin rotation algorithms. To accommodate these design variables, we extended the standard M2PL model to include correlated major factors and uncorrelated minor factors (to represent model error). Our results showed that both rotation methods converged to LS under some conditions with geomin producing the highest local solution rates across many models. Our results also showed that, for identical item response patterns, rotation LS can produce different latent trait estimates with different levels of measurement precision (as indexed by the conditional standard error of measurement). Follow-up analyses revealed that when rotation algorithms converged to multiple solutions, quantitative indices of structural fit, such as numerical measures of simple structure, will often misidentify the rotation that is closest in mean-squared error to the factor pattern (or item-slope pattern) of the data-generating model.