Classical pivots have frictional losses leading to the limited quality factor of oscillators used as time bases in mechanical watches. Flexure pivots address these issues by greatly reducing friction. However, they have drawbacks such as gravity sensitivity and limited angular stroke. This paper analyses these problems for the cross-spring flexure pivot and presents an improved version addressing these issues. We first show that the cross spring pivot cannot be both insensitive to gravity and have a long stroke. A 10 ppm sensitivity to gravity acceptable for watchmaking applications occurs only when the leaf springs cross at about 87.3 % of their length, but the stroke is only 30.88 % of the stroke of the symmetrical cross-spring pivot. For the symmetrical pivot, gravity sensitivity is of the order of 104 ppm. Our solution is to introduce the co-differential concept which we show to be gravity insensitive. We then use the co-differential to build a gravity insensitive flexure pivot with long stroke. The design consists of a main rigid body, two co-differentials and a torsional beam. We show that our pivot is gravity insensitive and achieves 100 % of the stroke of symmetrical pivots.