We experimentally investigate the presence of anisotropy from the inertial to the dissipative scales in homogeneous turbulence. We employ an apparatus in which two facing arrays of randomly actuated air jets generate turbulence with negligible mean flow and shear, over a volume several times larger than the energy-containing eddy size. The Reynolds number based on the Taylor microscale is varied in the range Reλ ≈ 300-500, while the axial-to-radial ratio of the root mean square velocity fluctuations ranges between 1.37 and 1.72. Two velocity components are measured by particle image velocimetry (PIV) capturing from the inertial to the Kolmogorov scales and yielding statistics up to sixth order. The scaling exponents of the velocity structure functions are found to differ not only between the longitudinal and transverse components, but also between the axial and radial directions of separation. At the dissipative scales, the moments of the velocity gradients indicate that departure from isotropy is, at the present Reynolds numbers, significant and more pronounced for stronger large-scale anisotropy. The structure-function based measures of isotropy tend towards isotropy as the separation is reduced from the inertial to the near-dissipative scales (down to about 10η, η being the Kolmogorov length) but become more anisotropic for even smaller scales which are characterized by high intermittency.