This paper proposes a new learning approach for evolving dynamic gaits of a hexapod robot. The controller that coordinates the leg movements consists of fully connected recurrent neural networks (FCRNNs). To automate the FCRNN parameter design, a symbiotic species-based particle swarm optimization (SSPSO) algorithm is proposed. There are multiple swarms in the SSPSO, where a swarm only optimizes the relevant parameters to a single node. The number of swarms is equal to the number of nodes in an FCRNN. The symbiotic behavior of particles from different swarms corresponds to the symbiotic structure of different nodes in an FCRNN. For a particle update, particles in different swarms update independently using a local version of particle swarm optimization (PSO) based on speciation. In each swarm, species are formed adaptively in each iteration according to both particle distance and performance. The design of FCRNNs using the SSPSO for temporal sequence generation and hexapod robot dynamic gait evolution for forward movement is conducted. For the latter, a multiple-FCRNN controller is first designed using a simulated hexapod robot. The designed controller is then successfully applied to a real hexapod robot gait control. The SSPSO is compared with the genetic algorithm and different PSO algorithms to verify its efficiency and effectiveness.