This paper investigates communication-efficient neural network transmission by exploiting structured symmetry constraints in convolutional kernels. Instead of transmitting all model parameters, we propose a degrees-of-freedom (DoF) based codec that sends only the unique coefficients implied by a chosen symmetry group, enabling deterministic reconstruction of the full weight tensor at the receiver. The proposed framework is evaluated under quantization and noisy channel conditions across multiple symmetry patterns, signal-to-noise ratios, and bit-widths. To improve robustness against transmission impairments, a projection step is further applied at the receiver to enforce consistency with the symmetry-invariant subspace, effectively denoising corrupted parameters. Experimental results on MNIST and CIFAR-10 using a DeepCNN architecture demonstrate that DoF-based transmission achieves substantial bandwidth reduction while preserving significantly higher accuracy than pruning-based baselines, which often suffer catastrophic degradation. Among the tested symmetries, \textit{central-skew symmetry} consistently provides the best accuracy-compression tradeoff, confirming that structured redundancy can be leveraged for reliable and efficient neural model delivery over constrained links.