= Solution
The coordinate axes are exactly the simultaneous eigenspaces of all invertible diagonal <matrices> in $T$. A <matrix> normalizing $T$ must permute these common eigenspaces: if $tv=\chi(t)v$, then $t(gv)=g(g^{-1}tg)v$ is again a scalar multiple of $gv$. Thus a normalizer element has exactly one nonzero entry in each row and column, a <monomial matrix>. In dimension two it is either diagonal or diagonal times $J=\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)$. Conversely $J$ swaps the diagonal entries and normalizes $T$. Therefore
$$
\boxed{N=T\rtimes\langle J\rangle=T\cup TJ,\qquad N/T\cong C_2.}
$$
This is the <normalizer of a diagonal subgroup of GL2>; the quotient records the permutation of the two axes.
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