The ability to rekindle a glowing splint is an indicator of the presence of an oxidizing agent, typically oxygen or a substance that releases oxygen. Among oxides of nitrogen, only a few are capable of doing this.
Nitrogen(I) oxide, commonly known as nitrous oxide (N2O), is not a strong enough oxidizer to rekindle a glowing splint.
Nitrogen(II) oxide, known as nitric oxide (NO), is not stable in the presence of oxygen and does not have the ability to rekindle a glowing splint because it does not actively release oxygen.
Nitrogen(IV) oxide or nitrogen dioxide (NO2), can support combustion by releasing oxygen as it decomposes. It is a brown gas and an effective oxidizer.
Dinitrogen tetraoxide (N2O4) is in equilibrium with nitrogen dioxide (NO2). However, at standard conditions, it is not as effective an oxidizer for rekindling a glowing splint as pure NO2.
In conclusion, the oxide of nitrogen that can rekindle a glowing splint is nitrogen(IV) oxide or nitrogen dioxide (NO2) due to its ability to release oxygen and support combustion.