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Existing observations of Uranus and Neptune's fundamental physical properties can be fitted with a wide range of interior models. A key parameter in these models is the bulk rock:ice ratio and models broadly fall into ice-dominated (ice giant) and rock-dominated (rock giant) categories. Here we consider how observations of Neptune's atmospheric temperature and composition (H2, He, D/H, CO, CH4, H2O and CS) can provide further constraints. The tropospheric CO profile in particular is highly diagnostic of interior ice content, but is also controversial, with deep values ranging from zero to 0.5 parts per million. Most existing CO profiles imply extreme O/H enrichments of >250 times solar composition, thus favouring an ice giant. However, such high O/H enrichment is not consistent with D/H observations for a fully mixed and equilibrated Neptune. CO and D/H measurements can be reconciled if there is incomplete interior mixing (ice giant) or if tropospheric CO has a solely external source and only exists in the upper troposphere (rock giant). An interior with more rock than ice is also more compatible with likely outer solar system ice sources. We primarily consider Neptune, but similar arguments apply to Uranus, which has comparable C/H and D/H enrichment, but no observed tropospheric CO. While both ice-and rock-dominated models are viable, we suggest a rock giant provides a more consistent match to available atmospheric observations. [ABSTRACT FROM AUTHOR]