omega plagued by Tissot or vice versa

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There are some old Tissot and Omega bipedal movements, often described as equivalent or similar. What really is the case? Has Omega developed the movement and Tissot built them under a different name? Or did Tissot invented Omega and use them in their watches too? Examples: Tissot 28 and Omega 340.
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In 1925 the Omega company appointed Paul Tissot as their commercial director. In 1929 it was decided to merge the two businesses; this was done in 1930 with the establishment of the Societe Suisse pour l'lndustrie Horlogere (SSIH), headquartered in Geneva.
The Tissot brand from then on covered the medium-price segment of the group's models also providing some movements to Omega