Abstract <p>We use an original approach to establish new Fejér type integral inequalities. This leads to two notable innovations: a relaxation of the classical but restrictive ‘‘symmetry with respect to a given point’’ assumption on the weighting function for the upper bound, and a new form for the lower bound. Our approach is based on an intermediate convexity result, a well-configured change of variables depending on the primitive of the weighting function, and an appropriate use of the famous Hermite–Hadamard integral inequalities. It supports the consideration of a flexible differentiable non-increasing assumption, which is still much more common in analysis than the symmetry assumption. Some refinements of the main result are also proposed, offering a new perspective on a classical mathematical topic. A numerical example illustrates the theory.</p>