Peter Bergmann (a student of Einsteins) in THE RIDDLE OF GRAVITATION, PAGE 60 notes

And here I had been thinking only about the speed of light as invarient.
Is the above coincidence?? I can't tell....

and the space time interval, is also invarient, right??

Are there other entities constant for all observers? Yes, I think four vectors such as "four vector momentum" which in simple terms is E^{2} - p^{2} that is (energy)^{2} - (momentum)^{2}...

I realize these transform (from one reference frame to another) as they do because of the mathematics of the formulations themselves, and the way they fit into relativity, but I can't help wondering if there is any deeper explanation or understanding. Any ideas?

The norm of four-momentum that you mention is the invariant mass (aka rest mass) which is the same for all observers.

I don't know about charge, I would guess that you could set up a current vector and then charge would be the timelike component, but then that wouldn't go along with Bergmann's comment. There must be some other four-vector that has charge as the norm. I actually haven't done much in the way of relativistic EM.

This is how I understand it (but I'm not sure at all to be correct): take a scalar quantity which depends on the frame of reference, e.g. energy, time difference between events, frequency, ecc., then take a vectorial quantity which depends on the frame of reference and that is related with the first (there exist an equation which involve both) example: energy and momentum or time difference and spatial distance, frequency and wave vector, ecc. Then there must exist a four-vector made with that scalar and that vectorial quantity, which square modulus is invariant, because for the relativity principle, all frames of reference must be equivalent, so the above equation must be valid in every of them. All 4-vectors have this property.

Example: if in a frame of ref. S you find: (c*delta t)^2 = 1.5 + (delta x)^2 + (delta y)^2 + (delta z)^2 then you also have to find: (c*delta t')^2 = 1.5 + (delta x')^2 + (delta y')^2 + (delta z')^2 in another frame of ref. S' , so if you write (c*delta t)^2 - (delta x)^2 + (delta y)^2 + (delta z)^2 you have found an invariant quantity because it's always = 1.5

An intriguing (for me) invariant quantity is Phase. I'd like to know which is the 4-vector which square modulus is the phase.