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 LK∗0K¯∗0 observable that compares the longitudinal components of both 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. 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 potential New Physics explanations for this deviation 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 possible anomaly.