Answer: If you play a fifth, the lower tone is the "root" of the implied chord. For a fourth, the upper tone is the root. The Pink Panther is in E minor, and those opening notes as written by Henry Mancini are fifths sliding from C#-G# up to E-B, suggesting a root position E minor chord.

If you played the same tones but inverted, i.e. G#-C# sliding up to B-E, it would be the same harmony, but with different chord positions. I think it sounds good that way, too, though beginning the piece with a 2nd-inversion chord might seem a little odd. Still, many people wouldn't notice the difference because the harmony remains.

In general you can change the position of any chord in music, and the effect of the change will not profoundly alter the piece except in certain cases where the identity of the root or the voicing of the chord is especially important and part of the drama, so to speak. But you do need to keep the same harmony.

So, the answer to your question is: no, intervals are named just according to the distance between the tones and the number of scale tones included. From any flavor of C rising to any flavor of G would be a fifth, because it covers 5 letters. But C going down to G covers four letters, so it's a fourth, and so is G ascending to C. In this case the most important thing is to start with a C# and a G#. Mancini would probably prefer you to have the C# underneath, but you could get away with reversing the two as long as the pitches are the same.