The cause is the way the LCD panel itself is made. Short of re-orienting the monitor, there's no way to alter that behavior. The "bottom" is indeed typically chosen as the "bad" angle to view from, on the assumption that normally the monitor will be oriented such that it's impossible to really view it from a very steep angle below (something else, like a table, would be in the way). Rotating it 90 degrees, however, puts the "bottom" off to one side, and you get the problem you describe.
What you've described is pretty typical of S-PVA panels commonly used on mid-high end, larger format monitors. An S-IPS/H-IPS panel shouldn't do that, and they are made in 19" (generally 5:4, not 4:3). Unfortunately, nobody publishes the actual specs of their monitors; often several different panels are used during the lifespan of the same model "monitor", so you get to play the panel lottery. You may be able to find an old PC monitor with an S-IPS panel that would work better for this application. They'll still get a little dimmer when viewed from "below", but they won't exhibit color shift even at pretty extreme angles.
Note that a lot of cheap consumer PC LCD monitors are REALLY bad in this regard. On many of them, even in their natural orientation, just looking 5 degrees off to either side results in noticeable color shift. This is typical of the cheap (but fast, giving a good figure for that "response time" figure people put way too much stock in) TN type panel.