The region I actually wanted to visualise in case of the second example was the area between y=4-x^2-z^2, x+z = 2 and the three coordinate planes. I figured that renaming the axes was a first good step towards a decent figure.

(The first example was the region between y=z^2 and z=y^2, x=0 and x+y+z=2; as somebody pointed out, I could make use of the symmetry there, but I was hoping there was some kind of generic command to turn a right-handed axes system into a left-handed one and vice versa)

Oops! I pasted the wrong example.
Anyhow, I have two examples where this goed utterly wrong. The full code is:

with(plots): with(plottools):A := shadebetween(0, 2-x-y, x = 0 .. 1, y = x^2 .. sqrt(x), scaling = constrained, color = yellow, axes = normal, labels = [Z, X, Y], transparency = .9); reflect(A, [[0, 0, 0], [1, 1, 0], [0, 0, 1]])

If I try it out, the labels on the axes disappear;

and

with(plots): with(plottools):A := shadebetween(0, -x^2-y^2+4, x = 0 .. 2, y = 2-x .. sqrt(-x^2+4), scaling = unconstrained, color = yellow, axes = normal, labels = [X, Z, Y], transparency = .9); reflect(A, [[0, 0, 0], [1, 1, 0], [0, 0, 1]])

Here the label on Y disappears, but judging from the fact that the Y-axis is the only one without a label, I still get a left handed set of axes.

Sorry for the confusion,

Still does not give the desired result... I've tried

A := shadebetween(0, -x^2-y^2+4, x = 0 .. 2, y = 2-x .. sqrt(-x^2+4), scaling = unconstrained, color = yellow, axes = normal, labels = [X, Z, Y], transparency = .9); reflect(A, [[0, 0, 0], [1, 1, 0], [0, 0, 1]])

Yes, I saw that thread but it doesn't seem to work.

A := shadebetween(0, -x^2-y^2+4, x = 0 .. 2, y = 2-x .. sqrt(-x^2+4), scaling = unconstrained, color = yellow, axes = normal, labels = [X, Z, Y], transparency = .9); reflect(A, [[0, 0, 0], [1, 0, 0], [0, 1, 0]])

gives:

Error, (in geom3d:-reflect) wrong number of arguments

Looks nice enough for me... what I didn't know was that I could make the border values of y within a plot command dependent of the values of x.

Just a note to all the helpful folks that I've finally setteled for the following:

A := plot3d([r*cos(phi), r*sin(phi), f(r, phi)], r = 3 .. 4, phi = 0 .. 2*Pi, color = yellow, scaling = constrained, style = patchnogrid, view = [-5 .. 5, -5 .. 5, 0 .. 5], axes = normal, labels = [x, y, z]);
B := plot3d(5, phi = -(1/2)*Pi .. (1/2)*Pi, theta = 0 .. 2*Pi, coords = spherical, numpoints = 10000, style = patchnogrid, color = yellow, scaling = constrained, axes = normal, labels = [X, Y, Z], transparency = .6);
C1 := implicitplot3d(x^2+y^2 = 9, x = -5 .. 5, y = -5 .. 5, z = 0 .. 4, numpoints = 10000, style = patchnogrid, color = yellow, scaling = constrained, axes = normal, labels = [X, Y, Z], transparency = 0);
C2 := implicitplot3d(x^2+y^2 = 16, x = -5 .. 5, y = -5 .. 5, z = 0 .. 3, numpoints = 10000, style = patchnogrid, color = yellow, scaling = constrained, axes = normal, labels = [X, Y, Z], transparency = 0);
display({A, B, C1, C2})

I'm kind of a newbie here, I don't seem to be able to plot the result here, but it's exactly what I've been looking for!

@Kitonum Nice! That was what I was looking for...