The hyperbola $y^2-x^2=1$ when rotated 45 degrees about the origin assumes the pretty simple form $y=1/x.$
What is the explicit cartesian equation for y=1/x revolved about y=x?
After a lot of work I arrived at $xy-1=z^2$ but I'm not sure if this is right. I do know that this is a two sheeted hyperbola.
On
First of all (as @Trebor has remarked), the point of intersection $I$ of hyperbola $xy=1$ with line $y=x$ being $(1,1)$, we have distance $OI=\sqrt{2}$.
You have recognized that you must have a hyperboloid with 2 sheets, with no values for $|z|<\sqrt{2}$, due to distance $OI$. Moreover the contour lines should be circles.
Equation $xy-1=z^2$ cannot be the correct one because for any $z=z_0$, we get a horizontal contour line $y=\dfrac{1+z_0^2}{x}$ which is a hyperbola and not a circle.
In order to have circles, the equation should have the form
$$x^2+y^2=R^2 \ \ \text{with} \ \ R^2=z^2-2\tag{1}$$
in order to prevent $R$ to exist if $z \in (-\sqrt{2},+\sqrt{2})\tag{2}$.
Gathering (1) and (2):
$$x^2+y^2-z^2=-2\tag{3}$$ is the equation of a hyperboloid with 2 sheets.
Let $a$ be any number and $b=\sqrt{a^2+1}$. The points $(b-a,b+a)$ lie on the hyperbola $xy=1$. The circle has radius $a\sqrt2$, so its coordinates are $$(b-a\sin\theta, b+a\sin\theta, a\sqrt2\cos\theta)$$
So $$(x-y)^2+2z^2=4a^2=(x+y)^2-4\\ z^2=2xy-2$$