I already asked this question here at mathoverflow but now that it seems to be clearer that the definition of a twisted product given below is indeed the one meant in the overview paper, the questions fits here better:
In the paper Multiply Twisted Products by Yong Wang, general definitions for so called warped and twisted products are given:
A (singly) warped product $B \times_b F$ of two pseudo-Riemannian manifolds $\left(B\,,g_B\right)$ and $\left(F\,,g_F\right)$ with a smooth function $b\,:\,B \to \left(0\,, \infty \right)$ is the product manifold $B \times F$ with the metric tensor $g = g_B \oplus b^2 g_F$. We call $\left(B\,,g_B\right)$ the base manifold, $\left(F\,,g_F\right)$ the fiber manifold and $b$ the warping function.
A twisted product $\left(M\,,g\right)$ is a product manifold $M = B \times_b F$, with a smooth function $b\,:\,B \times F \to \left(0\,, \infty \right)$ and the metric tensor $g = g_B \oplus b^2 g_F$.
It seems that those are the definitions meant in this overview paper about the Kerr/CFT correspondence about the following metric
\begin{align} \mathrm{d}s^2 = J\left(1+\cos^2\theta\right)\left[-r^2dt^2 + \frac{dr^2}{r^2} + d\theta^2 \right] + \frac{4 \sin^2\theta}{1+\cos^2\theta} \left(d\phi + rdt\right)^2\,, \end{align}
which is supposed to be a " warped and twisted product of $AdS_2 \times S^2$ " (J is just scaling the metric). I think the author just meant twisted, because in the above definitions the twisted productis the generalization of the warped product. Unfortunately I can't neither make out the above metric to be $AdS_2 \times S^2$, nor that it is for $\theta = \frac{\pi}{2}$ a "twisted product of $AdS_2$ and a circle of constant radius" (as this paper suggests), as the suggested geometry should not have off-diagonal terms like $\mathrm{d}\phi \mathrm{d}t$ in the metric. For clarity, the metric of $AdS_2$ in Poincare coordinates is \begin{align} \label{eq:poincarepatch} ds^2 = - r^2dt^2 + \frac{dr^2}{r^2}\,, \end{align} where we set the curvature constant to $1$ to match the above expression.
How is the above twisted $AdS_2 \times S^2$? The coordinate transformation would have to get rid of the off-diagonal term $\mathrm{d}t\mathrm{d}\phi$.
Is there at least a theorem that ensures the existence of such a coordinate transformation?