There is no second inversion. It's not like the brain has to receive the image "right-side-up" ... or like one stage of the processing has to "invert" the image for the next stage.
From the moment the brain is born, from the very first images it sees, it is born processing an inverted image from the eyes and develops an understanding of what "up" means from those images. It does not have to do some extra processing to "invert" that image.
The left-right inversion is far more interesting. (Objects on the right side of your visual field are being imaged on the left side of both retinas). This is a different question from up-down inversion because you have a left and a right eye. Many people think that the right eye is processed by the left brain and the left eye is processed by the right brain. But what actually happens is more interesting ... the left side of both retinas goes to the left brain, and the right side of both retinas goes to the right brain! This is necessary for stereopsis ... comparing the images from left and right eye so that we perceive *distance* to objects. It turns out that predators have this configuration, while herbivores, which have eyes on the side of their heads instead of pointing forward, have brains wired more for independent processing for the two eyes (knowing the exact distance to an object is not as important has being able to see thing practically behind you).
But notice again, that there is no second inversion. Objects on the right side of your visual field are seen by the left brain, and vice versa. There is no need to invert the image (swap the image left-to-right) which in this case would involve swapping the information across the two hemispheres of the brain.