Spherical Harmonics Parity

In Mathematica’s documentation, the Spherical Harmonics are said to be defined as follows, for $l geq 0$:

Furthermore, we know that $cos(x)=cos(-x)$, hence one can be led to believe that $Y_l^m(-theta,phi)=Y_l^m(theta,phi)$.

A quick check with mathematica shows us that might not be the case

Table[Table[{SphericalHarmonicY[l, m, -([Pi]/2), 0], 
   SphericalHarmonicY[l, m, [Pi]/2, 0]}, {m, -l, l}], {l, 0, 2}]

as the $l=1=m$ values differ.

Am I doing something wrong? Does the value of $theta$ need to be in the canonical range $[0,pi]$. If so, how do I relate $-frac{pi}{2}$ to something in that range. The formulae I know allow me to relate it to $frac{3pi}{2}$, which is still outside the range.

SphericalHarmonicY [1,1,phi,theta] == ... LegendreP[1,1,Cos[phi]] ..

LegendreP[1, 1, x] == -Sqrt[1 - x^2]

LegendreP[1,1,Cos[phi]] ==  -Sqrt[1 - Cos[phi]^2] ==  -Sqrt[Sin[phi]^2] == -Sin[phi]