Z° Isospin: Understanding T³=0 and Weak Isospin in Particle Physics

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Z° and weak isopsin

The Z° is a mixture of the B (iso-scalar) and the W° (iso-triplet with T³=0). Is it OK to say that the Z° still has T³=0? Or is this meaningless due to the breaking of the original SUwI(2) * UY(1) to Uem(1) and the mixing in the Z°?
 
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Since both the iso-scalar and the T3=0 component of the iso-triplet have T3=0, after mixing the resulting states (Z0 and photon) should still have T3=0 (I think).
 
I think so, too, but is this really correct?
 
The Z is in an eigenstate of Q, and in an eigenstate of Y, right? So it must be in an eigenstate of T3.
 
right
 
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