It is shown that the expressions for the correlation tensors of homogeneous axisymmetric turbulence can be considerably simplifed compared to previous analyses of Batchelor (1946) and Chandrasekhar (1950). Representations of the axisymmetric two-point correlations tensors are found, such that each measurable correlation corresponds to a single scalar function, and moreover such that the equations of continuity relating different tensor components to each other take the most simple form. Reflectional symmetry in planes normal to but not in planes through the axis of symmetry is demanded, which allows a full description of states with rotation about the axis of symmetry. The second and third-order velocity correlation tensors and the first-order pressure velocity correlation tensor are analysed with the new method. Small separation expansions of the correlation functions yield the quantities which have to be measured to determine various terms in the governing equations for the Reynolds stresses and the dissipation tensor. A scalar Poisson equation for the pressure-strain is derived. and the single-pint solution is written as a sum of integrals over measurable correlation functions. The simplified analysis can be of great experimental importance. It reveals in a simple way how a full experimental picture of homogeneous axisymmetric turbulence can be obtained by measuring components of the velocity at two points at variable distance from each other on a line perpendicular to the mean flow in a wind tunnel. By using the Fouricr Besscl transform it is also shown that the three-demensional energy, transfer, and pressure stain spectra can be extracted from such measurements.