The main definitions of clench are (1) to close tightly, and (2) to grasp or grip tightly. It’s also a noun referring to a tight grip or grasp. The original definition of clinch is to fix or secure (as a nail or bolt) by bending down or flattening the end that protrudes. This definition gave rise to the now more common sense of clinch—to settle definitely and conclusively. This sense is often used in sports, where a berth or title is clinched when the team or competitor secures it.
Some dictionaries list clench and clinch as variants of each other, but this is just because the words have been confused so often for so long. Edited publications and careful writers keep them separate.
Examples
Don’t consciously try to jab one finger or clench your fist with your thumb on top. [Forbes]
Daimler AG and Rolls-Royce Group PLC clinched a $4.8 billion takeover of German enginemaker Tognum AG after raising their bid. [Edmonton Journal]
As my body trails behind, arms elongated and tendons stretched, my hands stubbornly clench the handles … [The Australian]
Michael Etienne, an attorney, clinched the run-off race for North Miami city clerk Tuesday. [Miami Herald]
Alas, I was never a particularly gifted performer and suffered from sphincter-clenching stage fright. [Guardian]
Darren Clarke ended a three-year wait for a win by clinching the Iberdrola Open in Majorca. [Scottish Daily Record]