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We investigate the ordering properties of vertically-vibrated monolayers of granular cylinders in a circular container at high packing fraction. In line with previous works by other groups, we identify liquid-crystalline ordering behaviour similar to that of two-dimensional hard rectangular particles subject to thermal equilibrium fluctuations. However, due to dissipation, there is a much stronger tendency for particles to cluster into parallel arrangements in the granular system. These clusters behave as a polydisperse mixture of long life-time 'superparticles', and some aspects of the system behaviour can be understood by applying mean-field theories for equilibrium hard rectangles, based on two-body correlations, to these 'superparticles'. Many other features of the granular system are different: (i) for small particle length-to-breadth ratio kappa, we identify tetratic ordering at moderate packing fractions and smectic fluctuations at higher packing fractions, with no sharp transition between the two states. Both types of ordering can be explained in terms of clustering. (ii) For large k, strong clustering precludes the stabilisation of a uniaxial nematic state, and the system exhibits a mixture of randomly-oriented clusters which, as packing fraction is increased, develops into states with smectic fluctuations, again through a diffuse transition. (iii) Vorticity excitations of the velocity field compete with smectic ordering, causing dynamic fluctuations and the absence of steady states at high densities; the tetratic state, by contrast, is very stiff against vorticity, and long-standing steady states, spatially and orientationally homogeneous except for four symmetrical defects located close to the wall, can be observed.