DFT-like transform using triangle waves instead of sin waves
The closest orthogonal transform I know of that might meet your needs is the Slant Transform. It's based on sawtooth(ish) waves, but some of the basis functions do resemble triangle waves:



(source: Applied Fourier transform)

It was developed for image coding/compression, but it seems like a reasonable first approach for the analysis of long-term linear trends/reversals in financial data. It doesn't seem like many of the key papers describing the transform are available [for free] online, but the following paper probably has sufficient detail to implement something:


  A Truncation Method for Computing Slant Transforms with Applications
  to Image Processing. M. M. Anguh, R. R. Martin. IEEE Trans.
  Communications 43 (6), 2103-2110, 1995. (author link) (pdf link)


Specifically, see Section III which gives the recursion relations used to construct the transform matrix.