A more geometric proof of David Speyer's argument would be the following: First note that a diffeo with orthogonal Jacobian preserves the length of curves, in particular of straight lines, thus it preserves the Euclidean distance. Then use the result that a map in $R^n$ that preserves Euclidean distance is the composition of a translation with an orthogonal linear map.

A geometric proof would be the following: First note that a diffeo with orthogonal Jacobian preserves the length of curves, in particular of straight lines, thus it preserves the Euclidean distance.

From this it follows easily that it maps lines to lines. From the fundamental theorem of affine geometry, it then follows that the diffeo is an affine map, whose linear part is clearly be orthogonal. (Or just use at once the result that a map in $R^n$ that preserves Euclidean distance is the composition of a translation with an orthogonal linear map.)