Rotational control over optically trapped particles has gained significant prominence in recent years. The marriage between light fields possessing optical angular momentum and the material properties of microparticles has been useful to controllably spin particles in liquid, air and vacuum. The rotational degree of freedom adds new functionality to optical traps: in addition to allowing fundamental tests of optical angular momentum, the transfer of spin angular momentum in particular can allow measurements of local viscosity and exert local stresses on cellular systems. We demonstrate optical trapping and controlled rotation of nanovaterite crystals. These particles represent the smallest birefringent crystals ever trapped and set into rotation. Rotation rates of up to 5kHz in water are recorded, representing the fastest rotation to date for dielectric particles in liquid. Laser-induced heating results in the superlinear behaviour of the rotation rate as a function of trap power. We study both the rotational and translational modes of trapped nanovaterite crystals. The particle temperatures derived from those two optomechanical modes are in good agreement, which is supported by a numerical model revealing that the observed heating is dominated by absorption of light by the particles rather than by the surrounding liquid. A comparison is performed with trapped silica particles of similar size. The use of nanovaterite particles open up new studies for levitated optomechanics in vacuum as well as microrheological properties of cells or biological media. Their size and low heating offers prospects of viscosity measurements in ultra-small volumes and potentially simpler uptake by cellular media.