Total energy is conserved because time is homogeneous (time translation invariance). Total linear momentum is conserved because space is homogeneous (space translation invariance). Total angular momentum is conserved because space is isotropic (rotational invariance). These are examples of how symmetries determine the laws of physics.
Another way of looking at it is that linear and rotational movement are relative. There is no absolute reference point or interval or angle.
The symmetry of space and time means that space and time are relative. The laws of physics should reflect this relativity as much as other relativities. Whether space or time are the independent variable should be irrelevant to the laws of physics.
So there should be a transformation of space into time and time into space that preserves the laws of physics, i.e., that is invariant. That transformation in physics is based on the speed of light because light provides an unchanging reference between space and time.
The transformation is this: x’ = ct and t’ = x/c. That is, length and duration may be interchanged with the speed of light as a conversion factor, and the laws of physics will remain unchanged. An example of this is the Lorentz transformation.
J.H. Field has an article on this in the American Journal of Physics, volume 69 (5), May 2001, entitled “Space-time exchange invariance: Special relativity as a symmetry principle.” The difference is that he doesn’t know how time could be 3-dimensional. Now that we see time is 3-dimensional, there is no problem in affirming the symmetry of space and time.