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For a great explanation, why the Maxwell relations are "just a sneaky way of saying that the mixed partial derivatives of the function U commute". see https://johncarlosbaez.wordpress.com/2012/01/19/classical-mechanics-versus-thermodynamics-part-1/
(The fact that partial derivatives commute is known as Schwarz' theorem (see https://en.wikipedia.org/wiki/Maxwell_relations . Schwarz' theorem is simply a way of stating "the fact that a function S doesn’t change when we go around a parallelogram" https://johncarlosbaez.wordpress.com/2012/01/23/classical-mechanics-versus-thermodynamics-part-2/)