It has been previously observed that spatial independent component analysis (ICA), if applied to data pooled in a particular way, may lessen the need for spatial alignment of scans in a functional neuroimaging study. In this paper we seek to determine analytically the conditions under which this observation is true, not only for spatial ICA, but also for temporal ICA and for principal component analysis (PCA). In each case we find conditions that the spatial alignment operator must satisfy to ensure invariance of the results. We illustrate our findings using functional magnetic-resonance imaging (fMRI) data. Our analysis is applicable to both inter-subject and intra-subject spatial normalization.