Is there a "clean" way of parametrizing the intersection of the sphere and saddle? I was playing around, trying to find a parametrization for the intersection of the two surfaces,
$$x^2+y^2+z^2=1$$
$$z=x^2-y^2,$$
but I wasn't able to get anything nice-looking. Any suggestions, or is this a hopeless endeavor?

EDIT: Thanks, Thomas Andrews. With the corrected formulas,
$$ a(t) = \sqrt{\frac{2}{1+\sqrt{1+4\cos^2 2t}}} $$
$$ (x(t),y(t),z(t)) = (a(t)\cos t, a(t)\sin t, a(t)^2 \cos2t), $$
we get a very nice picture!


$$x^2+y^2=1-z^2\\
x^2-y^2=z$$
So $x^2=\frac{1+z-z^2}{2}$ and $y^2=\frac{1-z-z^2}{2}$.
Now you just need to pick the range of values for $z$ where both $1-z-z^2$ and $1+z-z^2$ are positive. This is when $1-z^2\geq |z|$ or $1-3z^2+z^4\geq 0$ and $|z|\leq 1$. But $1-3z^2+z^4=\left(z^2-\frac{3}{2}\right)^2 -\frac{5}{4}$, so you need $$z^2\leq \frac{3-\sqrt{5}}{2} \text{ or } z^2\geq\frac{3+\sqrt{5}}{2}$$
The right side is irrelevant, since it is already greater than the known bound $1$. Also: $\frac{3-\sqrt{5}}{2}=\left(\frac{\sqrt{5}-1}{2}\right)^2$. So you get that $|z|\leq\frac{\sqrt{5}-1}{2}$ and:
$$(x,y,z)=\left(\pm\sqrt{\frac{1+z-z^2}{2}},\pm\sqrt{\frac{1-z-z^2}{2}},z\right)$$

To correct the nice parameterization by Servaes in his wrong but close answer, if $(x,y,z)=(a\cos t,a\sin t, z)$ then $z=a^2(\cos^2 t -\sin^2 t)=a^2\cos 2t$.
Then $1=x^2+y^2+z^2=a^2+a^4\cos^2 2t$ or $$a^2=\frac{-1\pm\sqrt{1+4\cos^2 2t}}{2\cos^2 2t}=\frac{2}{1\pm\sqrt{1+ 4\cos^2 2t}}$$
Since you need $a^2\geq 0$, you can eliminate the $-$ case from $\pm$ and you get:
$$a(t)=\sqrt{\frac{2}{1+\sqrt{1+4\cos^2 2t}}}$$
and $$(x,y,z)=(a(t)\cos(t),a(t)\sin(t),a(t)^2\cos(2t))$$
