neglecting some unimportant (in this context) real constants, the fourier transform of the square pulse is
sin(w*t0)/w = (exp(i*w*t0) - exp(-i*w*t0)/(2*i*w)
If you incorporate the minus sign and the i in the denominator into the exponenentials the result is
= (exp(i*w*t0 - i*pi/2) + exp(-i*w*t0 + i*pi/2)/(2*w)
and you can see that the phase shift of the positive frquency part and the phase shift of the negative fequency part have opposite signs. So comparing positive to negative trequency, the phase shifts have opposite signs and the phase shift is odd.
When you take the angle of sin(w*t0)/w, which is a real, even function, you just get 0 for the phase when the function is positive and pi for the phase when the function is negative. So it's basically looking at the sign of a real function and that function happens to be even. But that is much different from comparing the positive and negative frequency contributions as should be done.