https://www.vintagehamilton.com/articles/the-urban-myth-about-the-hamilton-otis-reverso

"For years, the myth about the Hamilton Otis Reverso were heard by many.  The urban myth was Hamilton copied the watch without permission from LeCoultre, were sued by Le Coultre and almost put out of business because of their flagrant patent infringement.

...

On or around April 27, 1938, Hamilton was granted permission to use their patent (patent #1,930,416) to make a reversable watch (internal memo).  Earlier on, on January 28, 1938, Hamilton Watch Co sent a letter to the casemaker of Schwab and Wuischpard (S&W) outlining what they wanted to produce with the conditions Lecoultre also presented and S&W could only make this watch for the Hamilton Watch Co.  

Hamilton named many of their watches after people.  This watch was thought of being called the “Reverso” or the “Duo Plex,” but was eventually called the “Otis,” after Otis Byran, who was the Vice president and former chief pilot of T.W.A. Otis was a friend and skeet shooting partner of Ross Atkinson Hamilton Vice President in charge of sales, and in 1938, the Reverso was named the “Otis” after his friend.

So there you have it my fellow watch collectors and friends, the myth is bogus and Hamilton did indeed have permission to use the Reverso Patent!"