Is there a name for a pair of lattices which have the property given in the title (up to a change of variable)? The following example of a pair captures the property mentioned above:

$$(i)\ 1 + 80q^3 + 270q^4 + 432q^5 + 960q^6 + 2160q^7 + 3240q^8 + 5360q^9 + 8640q^{10}+\dots$$

$$(ii)\ 1 + 270q^4 + 960q^6 + 3240q^8 + 8640q^{10} + 17790q^{12} + 25920q^{14} + 62910q^{16} + \dots$$

The second theta series is given by taking only the even coefficients from the first series. The source of these series are ten dimensional lattices known as $O_{10}$ and $(C6\times SU(4,2)):C2$ from Nebe and Sloane's database of lattices.

Another example is $E_7$ and its dual lattice, having theta series given below (in reverse order)

$1 + 56q^3 + 126q^4 + 576q^7 + 756q^8 + 1512q^{11} + 2072q^{12} + \dots $

$1 + 126q^4 + 756q^8 + 2072q^{12} + 4158q^{16} + 7560q^{20} + \dots$