"In doing so" vs "In so doing"

'In doing so', makes the verb (doing) the focus of the phrase. 'In so doing', is slightly reflexive, returning the emphasis back to what is being done. It is a finely nuanced distinction and the degree of formality in the latter phrase lends it to a negative connotation. "And in so doing the defendant caused....". Whereas the former phrase 'In doing so' is easier to link with a more positive action.

John Lawler wrote in a comment just above:

They're both fixed phrases. The one with so before doing is more complex syntactically, and therefore more formal. Nobody talks like this; this is very old-fashioned written legal language. There are no meaning differences between them; which one to write (if either) depends on how formal the author wishes to appear. In doing so is Solicitor grade, but in so doing is Barrister.

'In so doing...' or "In doing so..."? Both collocations are possible!

With 'in doing so'—which means "as a consequence of doing" and where "so" means "the aforementioned"—the action ('doing') is given more attention. (Actually, in time you affix ING on a verb, you stress the action.)

By inverting 'doing' and 'so' ( to ''in so doing' ), the emphasis now falls on WHAT is being done, where "so" means "in that manner". The focus shifts to the CONSEQUENCES of [taking] action, whether the mood expresses realis or irrealis (true or hypothetical).

This syntactic alteration commonly known as inversion is also a good illustration of what I have coined "penultimacy".

The difference is subtle; it is a finely nuanced distinction in register (degree of formality). Once you can grasp that aspect —and two or three other important factors* that underscore what makes English the language it has always been at the core—you will find yourself more easily leaping over the hurdles that keep you from advancing beyond intermediacy in communication and argumentation.