Is there a lens that would invert my vision? Would I be able to invert my vision by mounting a particular lens in front of each eye?
I am currently able to achieve this by mounting a right-angled triangular prism across both eyes. The specific rotation can be thought of as follows:

The effect is that things in the "bottom" of my visual field appear at the "top" and "upsidedown". For example, if I stand and move my head to look down, I see my legs apparently "in front of me" and my feet pointing "towards me".
The prism arrangement is heavy, uncomfortable (there is an edge with an acute angle against my forehead), and severely restricts my view (which can be fully occluded by an A3 sheet of cardboard held at arms length). This is a problem as the scientific value of this arrangement comes from being able to wear them for extended periods (e.g. a whole day, or 4 hours a day for a month).
Is there a lens that would have essentially the same inversion effect, without all of the problems in the previous sentence? Suggested solutions don't have to be inexpensive.
 A: Binoculars include a prism that invert the image as the lenses already do an inversion. The lens apparatus would be a bit heavy to use as an inversion mechanism, but the prisms (Porro prims for instance) could do the job. They can provide a decent field of view. Even if they take some space and weight, I doubt any other apparatus, such as a lens arrangement, could do better.
A: I think you should consider a catoptric or catadioptric system. The heart of the system is two parabolic mirrors facing each-other (can be light weight if you aren't overly concerned about a flawless image) , and they produce an inverted image.
Here is a diagram showing the light path of a Cassegrain telescope.  Obviously you would use different curvatures for both primary and secondary because your objective is not to magnify but to invert.

A: This outfit sells goggles that do that. It's just a matter of prisms and some engineering.
Alternatively, you could just have them look down/up into a mirror, or even make a periscope like this, where the blue parts are mirrors:

or this:

A: This a very old question and it's practical application is related to neuroscience. If you read this Wikipedia article: Perceptual adaptation - Experimental support you will learn of 1890s experiments with inverted vision glasses aimed to verify that brain will adapt and flip the images at some point after long glasses usage. Here is description of glasses that were used by the experimentator George M. Stratton found here on 1st page:

This was done by binding on the eyes a simple optical contrivance
  constructed on the following principle : If two convex lenses of equal
  refractive power be placed in a tube at a distance from each other
  equal to the sum of their focal distances, the eye in looking through
  the tube sees all things inverted, but in other respects the image
  remains unchanged. The image cast on the retina is as if the whole
  field of view had been revolved on the line of sight through an angle
  of 180o. All light other than that which comes through the lenses
  must, of course, be carefully excluded by making the instrument fit
  exactly the inequalities of the face by means of black linings and
  pads. For if light were permitted to enter the eyes otherwise than
  through the lenses, the observer would be subjected to- both upright
  and inverted images, and the purity of the experiment would be lost.
  The size of the visual field was a matter requiring some care. The
  size and refractive power of the lenses are the determining factors
  here, and in the desire to obtain a reasonably large visual field one
  is tempted to use large thick lenses. But they are soon found to be
  too heavy to wear on the head for a considerable length of time. I
  found it best, therefore, to modify the instrument above described, by
  substituting two double convex lenses (placed close together on the
  same axis line) for each of the lenses in that description. I had thus
  for each eye a short adjustable tube, and at either end of the tube a
  pair of good lenses of equal focal length. The instrument by this
  means gave a clear field of vision with a compass of 45o, and at the
  same time was light enough to be worn without discomfort.

A: Two lenses placed such that the focal points of each overlap.
I'll give you an image to help...

The blue dot is the focal point of both lenses. The red and green paths are simple enough, and the brown shows how the angle of incoming light is flipped, so it looks like the light is coming in from the top, not the bottom.
BTW: I used what looks like to convex lenses, this could also be two flats. So even something as simple as a glass ball given the right index could do this.  
EDIT:
Reading this a second time over, makes me think you might want something different than two glass spheres. This would make things on the left appear to be on the right and things on your right, left. 
I think a glass cylinder lens would be more like it. That way the ray's horizontal angle remains unchanged. 
Are we trying to do this?
If so, you need only use two of these.
Like this.
Basically take two of those prisms you are using and sandwich them with aluminum or some other reflective material.
A: Andrew, do you have to have an optical solution ?   An electronic solution, combining a Google glass with an electronic image inversion would do the trick.  I know that HP made some small LED displays, that you wore like glasses, which projected a heads up display.  could easily be programmed to invert the image.
A: Not quite the type of solution you're looking for (and I know George E. Smith already suggested something similar), but lots of people have modified the Oculus Rift consumer virtual reality headset by mounting dual wide-angle webcams on the front, for augmented reality applications. It seems there are kits available for this.
This solution would have the advantages of being ergonomically designed and having a good field of vision and excellent isolation of light from outside the helmet. It would also be trivial to program it to flip horizontally rather than vertically, or to make other transformations to the visual field, if you need that.
On the other hand it does need to be plugged in all the time, and at 379 grams (according to Wikipedia) it might be a bit heavy and bulky for long-term wear. Still it seemed worth mentioning.
A: You can invert you sight using a dove prism.
