Q:

Where is the parabolic paraboloid? 

In theory, any quadric surface can, through a series of translations and linear transforms, be converted to $f(X)+g(Y)+h(Z)=C$, where $f(X)=X^2$ (except in the case of the imaginary ellipse, where $f(X)=-X^2$), $g(Y)$ and $h(Z)$ may be positive square, negative square, negative linear or zero functions, and $C$ is either $0$ or $1$. By my reckoning, doing these transforms would result in eleven distinct possible surfaces (counting cases that differ only by $C$'s value as not distinct).  These are:
$X^2+Y^2+Z^2=C$, the real ellipsoid (point if $C=0$)
$X^2+Y^2-Z^2=C$, the hyperboloid of one sheet (double cone if $C=0$) about the $Z$-axis
$X^2-Y^2-Z^2=C$, the hyperboloid of two sheets (double cone if $C=0$) about the $X$-axis
$-X^2-Y^2-Z^2=C$, the imaginary ellipsoid (point if $C=0$) 
$X^2+Y^2-Z=C$, the elliptic paraboloid
$X^2-Y^2-Z=C$, the hyperbolic paraboloid
$X^2-Y-Z=C$, which is unnamed but by analogy would be the parabolic paraboloid
$X^2+Y^2=C$, the elliptic cylinder (line if $C=0$)
$X^2-Y^2=C$, the hyperbolic cylinder (pair of intersecting planes if $C=0$)
$X^2-Y=C$, the parabolic cylinder and
$X^2=C$, the pair of parallel (coincident if $C=0$) planes. 
These aren't necessarily the canonical forms of course, but they do highlight an apparent gap in the literature.  I have yet to find any mention of a quadric surface with one square term and two linear terms, the parabolic paraboloid in the list.  Does it make sense to consider this surface?  Can it be reckoned in terms of the others?  What does it look like?  And where can I find literature on the subject? 

A:

$X^2-Y-Z=C$ (my "parabolic paraboloid") can be rotated into $X^2-Y=C$ (the parabolic cylinder). 
Thanks to Ofek Gillon for untangling the knot I had tied my brain into.