I would like to perform a principal component regression (PCR), but feel a little confused about the rotation type to be used in the principal component analysis (PCA) step. First I perform a PCA to extract components, but I observe cross loadings after the extraction. Therefore, I perform a rotation (orthogonal or oblique) to overcome the cross-loadings issue and make components more interpretable. The thing is PCR is performed to overcome multicollinearity. As far as I have learned, with an orthogonal rotation technique, I assure that the regressors in PCR (at this time the selected components) will not be correlated, which is something good to avoid multicollinearity. However, oblique rotation techniques allow components to be correlated each other. In this case is it wrong to perform PCR with an oblique rotation technique? In other words, can PCR be performed only with unrotated or orthogonally rotated components?
A really nice overview, put in layman's terms, of rotation in any sort of Exploratory Factor Analysis can be found here: What is Rotating in Exploratory Factors Analysis?
The short answer to your question is that both orthogonal and oblique rotations both produce (virtually) identical results in the highly unlikely case that factors are perfectly uncorrelated. Historically, researchers using EFA have been drawn to orthogonal rotations because uncorrelated factors are easier to interpret. However, if factors are correlated (such as in human psychology, where identifiable factor groupings are likely going to have some correlations), then we likely want our model to reflect that.
In using PCR, instead of EFA, you do not need clearly defined or interpret-able factors, as you are using the method as a predictive technique. If the world of covariates indicates that there is some correlation between your factors, you likely want to model that. However, PCA is not the preferred method for modelling, though it is great for data compression and representation. Try PLS if you are interested in the modeling ability of your covariates.