One of the books on "Conceptual Wavelets" by Fugal explains some major differences between the undecimated discrete wavelet transform (UDWT) vs. discrete wavelet transform (DWT). In UDWT the scale of wavelet is increased continuously just like the continuous wavelet transform, but the scale increases in dyads (powers of 2). In the DWT, which is the most commonly used in MATLAB, the filter size remains the same, but the data is reduced dyadically.
His exact wordings are "As discussed briefly in the preview, instead of dyadically stretching the filters, the conventional (decimated) DWT dyadically shrinks the signal instead"
He is using the example of Haar wavelets on a very small set of data, simply exam scores as a signal of eight exam scores [ 80 80 80 80 0 0 0 0].
So the question is when we downsample by 2, which one is more correct to throw away even samples or odd samples?