Polchinski String Theory (8.4.38) T-duality of two compactified dimensions

I am confused about a sentence in Polchinski’s String theory chapter 8 p 255 when he works out the example of the full $T$-duality with two compact dimensions. He writes

"A simultaneous $T$-duality on $X^{24,25}$ acts as $rhorightarrow -1/rho$ with $tau$ invariant."

Here $rho$ and $tau$ are two complex fields that contain the four moduli $G_{24,24}, G_{24,25}, G_{25,25}$ and $B_{24,25}$.

I am not able to derive this. What am I missing?

Details

The start is the non-linear model
$$
S= frac{1}{4pialpha’}int d^2sigma , sqrt{g} left[ left(g^{ab} G_{munu}(X) + i epsilon^{ab} B_{munu} right)partial_a X^mu partial_b X^nu + alpha’ R Phi(X) right]
$$
Introduce the moduli $rho$ and $tau$ in (8.4.36) and (8.4.37), i.e.
$$
G_{24,24} = frac{alpha’ rho_2}{R^2tau_2}; quad
G_{25,25} = frac{alpha’ rho_2 |tau|^2 }{R^2tau_2} ;quad
G_{24,25} = frac{ alpha’ rho_2tau_1}{R^2tau_2};quad
B_{24,25} = frac{alpha’rho_1}{R^2}
$$
with inverse
$$
rho_1 = frac{R^2}{alpha’} B_{24,25} ;quad
rho_2 = frac{R^2}{alpha’} sqrt{ G} ;quad
tau_1 = frac{G_{24,25}}{G_{24,24}};quad
tau_2 = frac{sqrt{ G}}{ G_{24,24}}
$$
where $G= det G_{mn}$.

A simultaneous $T$-duality on $X^{24,25}$ changes $Rlongrightarrow alpha’ /R$ and leaves the $X$ invariant. This implies that also the $G_{mn}$ and $B_{mn}$ are invariant (I think?) and thus
$$
rho_1 longrightarrow rho_1’= frac{alpha’^2}{R^4} rho_1;quad
rho_2 longrightarrow rho_2’= frac{alpha’^2}{R^4} rho_2
$$
and leaves $tau$ unchanged. But that does not correspond to $rholongrightarrow -1/rho$. What have I missed?