In the context of the recently measured non-leptonic decays Bd → K∗0K¯¯¯¯¯∗0 and Bs → K∗0K¯¯¯¯¯∗0 we analyse the anatomy of the LVV observable that compares the longitudinal components of Bs → VV and Bd → VV decays. This observable is cleaner than the longitudinal polarisation fraction as it is afflicted only at subleading order in a 1/mb expansion by the theoretical uncertainties arising in the transverse components entering the polarisation fraction. Focusing on the particular case of Bd → K∗0K¯¯¯¯¯∗0 and Bs → K∗0K¯¯¯¯¯∗0, we discuss the main sources of hadronic uncertainty in the SM. We find for the SM prediction LK∗K¯¯¯¯¯∗=19.5+9.3−6.8, which implies a 2.6σ tension with respect to the most recent data, pointing to a deficit in the b → s transition of the non-leptonic decay versus the corresponding b → d transition. We discuss possible New Physics explanations for this deviation, first at the level of the Weak Effective Theory and we identify that the two Wilson coefficients C4 and C8g can play a central role in explaining this anomaly. Finally, we briefly explore two different simplified New Physics models which can explain the anomaly through a contribution either in C4 (Kaluza-Klein gluon) or in C8g, with a significant amount of fine tuning, but possible connections to the b → sℓℓ anomalies.