Thanks to their narrow band gap nature and fairly high carrier mobility, HgTe nanocrystals are of utmost interest for optoelectronics beyond the telecom window (λ>1.55 µm). In particular, they offer an interesting cost-effective alternative to the well-developed InGaAs technology. However, in contrast to PbS, far less work has been dedicated to the integrating this material in photodiodes. In the shortwave infrared, HgTe NCs have a more p-type character than in the mid wave infrared, thus promoting the development of new electron transport layers with an optimized band alignment. As for perovskites, HgTe NCs present a fairly deep band gap with respect to vacuum. Thus, we were motivated by the strategy developed for perovskite solar cells, for which SnO2 has led to the best performing devices. Here, we explore the following stack made of SnO2/HgTe/Ag2Te in which the SnO2 and Ag2Te layers behave as electron and hole extractors respectively. Using X-ray photoemission, we show that SnO2 presents a nearly optimal band alignment with HgTe to efficiently filter the hole dark current while letting the photoelectrons flow. The obtained I-V curve exhibits an increased rectifying behavior, and the diode stack presents a high internal efficiency for the diode (above 60%) and an external quantum efficiency that is mostly limited by the absorption magnitude. Furthermore, we tackle a crucial challenge for the transfer of such diode onto readout circuits which prevents backside illumination. We also demonstrate that the diode stack is reversible with a partly transparent conducting electrode on top while preserving the device's responsivity. Finally, we show that such SnO2 layer is also beneficial for electron injection and leads to enhanced electroluminescent signal as the diode is operated under forward bias. This work is an essential step toward the design of a focal plane array with an HgTe NCs-based photodiode.