  We investigate few- and many-body states in half-filled maximally symmetric
topological insulator flat bands realized by two degenerate Landau levels which
experience opposite magnetic fields. This serves as a toy model of flat bands
in moir\'e materials in which valleys have Chern numbers $C= \pm 1$. We argue
that although the spontaneously polarized Ising Chern magnet is a natural
ground state for repulsive Coulomb interactions, it can be in reasonable
energetic competition with correlated states which can be viewed as Laughlin
states of excitons when short distance corrections to the interaction are
included. This is because charge neutral excitons in these bands behave
effectively as charged particles in ordinary Landau levels. In particular, the
Ising Chern magnet is no longer the ground state once the strength of a short
range intra-valley repulsion is about $30\%$ larger than the inter-valley
repulsion. Remarkably, these excitonic Laughlin states feature valley number
fractionalization but no charge fractionalization and a quantized charge Hall
conductivity identical to the Ising magnet, $\sigma_{xy}= \pm e^2/h$, and thus
cannot be distinguished from it by ordinary charge transport measurements. The
most compact excitonic Laughlin state that can be constructed in these bands is
an analogue of $\nu=1/4$ bosonic Laughlin state and has no valley polarization
even though it spontaneously breaks time reversal symmetry with a charge Hall
conductivity $\sigma_{xy}= \pm e^2/h$.
