What is the correct order of operations for a 2D Haar wavelet decomposition?
[Beginning of the story] Remember the discrete wavelet curse: in 1D, with 2-scale or dyadic wavelets, you cannot have finite support, realness, orthogonality and linear phase (symmetry/antisymmetry) at the same time, except for the Haar wavelet, which lacks of regularity and overlap. You have to lift one constraint to have the other fulfilled. For instance :


if you lift symmetry, you get Daubechies wavelets
if you lift the dyadic, 2-scale, you get FIR $M$-band filter banks, etc.


When extending discrete wavelets to 2D, many options appear. Genuine two-dimensional dyadic wavelets exist (e.g. Design of Regular Nonseparable Bi-dimensional Wavelets Using Gröbner Basis Techniques, 1998), but their lack for separability make them cumbersome (and not often used in practice). 

So, most people stick to the large background of known 1D dyadic wavelet designss and apply them on rows an columns of cartesian sampled data, like images. Thus is clearly a lack of imagination, and more genuine 2D, oriented, geometrical wavelets are possible, but the SE margin is too small to wrote about it.

Meanwhile, sticking to 1D wavelet designs, the order of operations on rows and columns matter. 

Two main schemes are classics, but depending on the literature, they are more or less known, and more or less used, and often under different names, so:


process all rows then all columns (or reverse): this scheme seems better known in numerical analysis and partial differential equations. It can be found under many names: separable, standard, S-form, rectangular, anisotropic, tensor, hyperbolic, separated.
process rows and columns alternatively: this scheme seems better known in image/video processing. It can be found under many names: non-separable wavelet transform, non-standard form, NS-form, square wavelet transform, isotropic wavelet transform, Mallat decomposition, isotropic wavelet transform, combined wavelet transform.


Some papers are given  here, and it is detailed in "Chapter 3. Oriented and geometrical multiscale representations" of the 2011 review paper on 2D wavelets: A panorama on Multiscale Geometric Representations, intertwining spatial, directional and frequency selectivity.

What is the best? I sure do not know, I I am still working on this. But it's geeting late, and I am attending these days a conference in honor of Alexandre Grossmann and Yves Meyer on wavelets in Paris, and I shall wake up early.

[EDIT: Added references to related questions]


2D DWT Image Issue
What is the correct order of operations for a 2D Haar wavelet decomposition?
2D DWT computation order


Let me backlet... rrr