We suggest symmetric variational physics-informed neural networks (symmetric VPINN) to learn the symmetric fluid flow and physical properties of fluids from a limited set of data. Symmetric VPINN is based on the VPINN framework and guarantees the symmetry of the solutions by modifying the network architecture. The effectiveness of the symmetric VPINN is demonstrated by predicting the velocity profiles and power-law fluid properties in the Poiseuille flow of a parallel channel. Symmetric VPINN models robustly and accurately learn power-law fluid flow in both forward and inverse problems. We demonstrate that the symmetric VPINN can be particularly useful when the power-law index is small and the data are extremely limited. The modified network architecture in the symmetric VPINN guides the neural network towards an exact solution by reinforcing symmetry. We show that symmetric VPINN is effective in obtaining unknown physical properties in practical experiments where data are scarce, suggesting the possibility of introducing known conditions of the system directly into the network structure to improve the accuracy of the network.