Answer
x-axis, y-axis, origin.
Work Step by Step
Step 1. To test symmetry with respect to the x-axis, replace $y$ with $-y$, we have $(-y)^2-(x)^2=-6$ or $(y)^2-(x)^2=-6$ which is the same the original, thus it is symmetric with respect to the x-axis.
Step 2. To test symmetry with respect to the y-axis, replace $x$ with $-x$, we have $(y)^2-(-x)^2=-6$ or $(y)^2-(x)^2=-6$ which is the same the original, thus it is symmetric with respect to the y-axis.
Step 3. To test symmetry with respect to the origin, replace $x,y$ with $-x,-y$, we have $(-y)^2-(-x)^2=-6$ or $(y)^2-(x)^2=-6$ which is the same the original, thus it is symmetric with respect to the origin.