I did some testing and it turns out that ITK allows reading sheared volumes (tested with nrrd and mha) but orthogonalizes axes when writing volumes (tested with mha, nrrd, nifti, mgz, mnc). This is a quite significant inconsistency.
Since ITK allows storage of sheared volumes in image data objects (source), we have two options:
Option A: enable sheared volumes in ITK image IO class inputs and outputs
Option B: not let ITK image IO classes to read/write sheared volumes (throw an error if such volumes are encountered)
As an application developer, I would prefer Option B, because it would allow our application to orthogonalize images (by resampling or refusing to read/write). As far as I know, this approach is used by VTK, too. An orthogonalization filter could be added to ITK (that would resample the sheared volume to have an orthogonal volume). In the future, it might be even implemented in image IO base class and performed automatically (could be enabled/disabled).
If there is an agreement that we should go towards option B, then a first step could be to remove orthogonalization during image writing for file formats that can store sheared volume and throw an error (instead of silent data corruption) in file formats that can only store orthogonal volumes.