Significance of sequency ordering in Walsh-Hadamard matrices So I've been doing some reading into Hadamard and related matrices (Slant, Walsh, , and have observed that there are three typical "orderings" of the rows based on various criterion. E.g. "Natural", "Dyadic", "Walsh" (or Sequency http://fourier.eng.hmc.edu/e161/lectures/wht/node3.html).

One of the books I'm reading is "Introduction to orthogonal transforms with applications in data processing and analysis" by Ruye Wang (of which a copy can be accessed freely here http://fourier.eng.hmc.edu/book/lectures/mybook.pdf )

In this book there is a chapter (8) dedicated to some of these matrices. In particular is the Hadamard matrix. It states a definition of sequency: 


  "..the sequency s of each row, defined as the number of
  zero-crossings or sign changes in the row. Similar to frequency,
  sequency also measures the rate of changes or variations in a signal.
  However, sequency can measure non-periodic signals as well as periodic
  ones"


Later on in the discussion of making the Hadamard matrix into a seqeuency ordered Walsh-Hadamard matrix it states:


  "The rows (or columns) of the WHT matrix H, ... , are not
  arranged in the order of the sequencies, while it is desireable to
  arrange them according to the sequencies in a low-to-high order,
  similar to how the DFT coefficients are arrange."


In the section on the Haar transform, it makes a brief explanation comparing the Fourier, Walsh-Hadamard, and Haar transforms:


  "It is interesting to compare the Haar transform with other orthogonal
  transforms such as the Fourier, cosine, Walsh-Hadamard, and slant
  transforms discussed before. What all of these transforms, as well as
  the Haar transform, have in common is that their coefficients
  represent some types of detail contained in the signal, in terms of
  different frequencies (Fourier transform and cosine transform),
  sequencies (Walsh-Hadamard transform), or scales (Haar transform), in
  the sense that more detailed information is represented by
  coefficients for higher frequencies, sequencies, or scales"


One can observe an analogy between the fourier and sequency ordering ( Aittokallio et al, 2001 - Testing  for  Periodicity  in  Signals: An Application  to  Detect  Partial  Upper Airway  Obstruction  During  Sleep:


  Walsh  functions with even  and  odd  orders  are  called the cal and
  sal functions,  respectively, and  they correspond to  the cosine and
  sine  functions in Fourier  analysis.  One should note, that in
  contrast to harmonic functions the


TLDR

Now, i've been seeing the Walsh-Hadamard matrix in various papers regarding locality sensitive hashing, and approximate randomization, random projections etc. Returning to the Walsh-Hadamard, my confusion is that I'm trying to form an intuition behind the significance of using a sequency ordered Hadamard matrix as opposed to just a standard Hadamard matrix, in general, and in instances when the WH matrix is combined with other matricies to induce randomization (e.g. see Subsampled Randomized Transform, e.g. Tropp 2010 - IMPROVED  ANALYSIS  OF  THE
SUBSAMPLED  RANDOMIZED  HADAMARD  TRANSFORM)
There isn't much significance. Except for few very particular applications like auto-focusing where you might try to adjust the lens such that the higher sequency components has maximum magnitude.  The FFT of a particular sequency isn't bandwith limited. Therefore sequency has very limited meaning.
You should indeed be looking at the applications like random projections, locality sensitive hashing, associative memory and neural networks:
WHT applications