Charge-carrier confinement in nanoscale In-rich agglomerations within a lateral InGaAs quantum well (QW) formed from stacked submonolayers (SMLs) of InAs in GaAs is studied. Low-temperature photoluminescence (PL) and magneto-PL clearly demonstrate strong vertical and weak lateral confinement, yielding two-dimensional (2D) excitons. In contrast, high-temperature (400 K) magneto-PL reveals excited states that fit a Fock-Darwin spectrum, characteristic of a zero-dimensional (0D) system in a magnetic field. This paradox is resolved by concluding that the system is heterodimensional: the light electrons extend over several In-rich agglomerations and see only the lateral InGaAs QW, i.e., are 2D, while the heavier holes are confined within the In-rich agglomerations, i.e., are 0D. This description is supported by single-particle effective-mass and eight-band k⋅p calculations. We suggest that the heterodimensional nature of nanoscale SML inclusions is fundamental to the ability of respective optoelectronic devices to operate efficiently and at high speed.