The invert elevation is the elevation of the inside bottom of a pipe or channel where it enters or leaves a structure. Water flows along the invert, so these numbers control the slope of the whole run and cannot be fudged.
On a precast structure, each pipe opening has to be placed at the right invert and the right angle. The difference between the rim and the lowest invert sets how tall the structure is. Miss an invert on the shop drawing and the piece will not line up with the pipe in the field.