  The layered honeycomb magnet $\alpha$-Li$_2$IrO$_3$ has been theoretically
proposed as a candidate to display novel magnetic behaviour associated with
Kitaev interactions between spin-orbit entangled $j_{\rm eff}=1/2$ magnetic
moments on a honeycomb lattice. Here we report single crystal magnetic resonant
x-ray diffraction combined with powder magnetic neutron diffraction to reveal
an incommensurate magnetic order in the honeycomb layers with Ir magnetic
moments counter-rotating on nearest-neighbour sites. This type of magnetic
structure has not been reported experimentally before in honeycomb magnets and
cannot be explained by a spin Hamiltonian with dominant isotropic (Heisenberg)
couplings. The magnetic structure shares many key features with the magnetic
order in the structural polytypes $\beta$ and $\gamma$-Li$_2$IrO$_3$,
understood theoretically to be stabilized by dominant Kitaev interactions
between Ir moments located on the vertices of three-dimensional hyperhoneycomb
and stripyhoneycomb lattices, respectively. Based on this analogy and a
theoretical soft-spin analysis of magnetic ground states for candidate spin
Hamiltonians, we propose that Kitaev interactions also dominate in
$\alpha$-Li$_2$IrO$_3$, indicative of universal Kitaev physics across all three
members of the harmonic honeycomb family of Li$_2$IrO$_3$ polytypes.
