We know do know:
   They were made in France.
   There was a Palmer razor that had the same or very similar head but the handle had diamond type knurling.
What we can assume is that between "le Coq" and "Palmer" they agreed to sell the razor under the Le Cog brand and that some sort of fee was agreed.
From the patent dates we also know that they were sold from the Mid 1930's very likely into the 1940's. At that time there were a few competing brands that were manufacturing adjustable razors in France.
I found an image of a pamphlet online showing their address as: Lames et Rasoirs Le Coq - 8, rue Revol, Parc-Saint-Maur (Siene).
I haven't been able to find more information and there is nothing to suggest that they were related to the company that does the sportswear.

Moving onto the "adjustability" part.
It has a spring in the centre that holds the tension on the blade. It works OK'ish but means that as you move towards the outer edges there is less and less blade support. Another anomaly with it is that it has no drainage so needs to be rinsed often.

As far as I can pick-up, there are four versions.
    The Plated Brass version.
    The Plated Brass version with a wide base plate. (See below - The photo is not mine - I don't have one in my collection)
    The Plated Zamak (Zinc) with a "small adjuster knob" version. I've seen the adjuster dial in black and brass
    The Plated Zamak (Zinc) with a "big adjuster knob" version.

All three are covered under patent number FR801804 which was granted on 19 August 1936 to MM Paul Baron and Henri E.G.J Morazin.

Le Coq Brass Version	Le Coq Zamak (Zinc) Version	Le Coq Big Adjustment Knob

This is the Brass Version with a wider base plate. Other than the wide base plate the rest of the razor is the same as the Brass Version (This is not mine. The image was sent to me by a fellow collector - Mike G)