Despite the widely acknowledged significance of turbulence-driven clustering, a clear topological definition of particle cluster in turbulent dispersed multiphase flows has been lacking. Here we introduce a definition of coherent cluster based on self-similarity, and apply it to distributions of heavy particles in direct numerical simulations of homogeneous isotropic turbulence, with and without gravitational acceleration. Clusters show self-similarity already at length scales larger than twice the Kolmogorov length, as indicated by the fractal nature of their surface and by the power-law decay of their size distribution. The size of the identified clusters extends to the integral scale, with average concentrations that depend on the Stokes number but not on the cluster dimension. Compared to non-clustered particles, coherent clusters show a stronger tendency to sample regions of high strain and low vorticity. Moreover, we find that the clusters align themselves with the local vorticity vector. In the presence of gravity, they tend to align themselves vertically and their fall speed is significantly different from the average settling velocity: for moderate fall speeds they experience stronger settling enhancement than non-clustered particles, while for large fall speeds they exhibit weakly reduced settling. The proposed approach for cluster identification leverages the Voronoï diagram method, but is also compatible with other tessellation techniques such as the classic box-counting method.