Humans have an exquisite sense of touch which robotic and prosthetic systems aim to recreate. We developed algorithms to create neuron-like (neuromorphic) spiking representations of texture that are invariant to the scanning speed and contact force applied in the sensing process. The spiking representations are based on mimicking activity from mechanoreceptors in human skin and further processing up to the brain. The neuromorphic encoding process transforms analog sensor readings into speed and force invariant spiking representations in three sequential stages: the force invariance module (in the analog domain), the spiking activity encoding module (transforms from analog to spiking domain), and the speed invariance module (in the spiking domain). The algorithms were tested on a tactile texture dataset collected in 15 speed-force conditions. An offline texture classification system built on the invariant representations has higher classification accuracy, improved computational efficiency, and increased capability to identify textures explored in novel speed-force conditions. The speed invariance algorithm was adapted to a real-time human-operated texture classification system. Similarly, the invariant representations improved classification accuracy, computational efficiency, and capability to identify textures explored in novel conditions. The invariant representation is even more crucial in this context due to human imprecision which seems to the classification system as a novel condition. These results demonstrate that invariant neuromorphic representations enable better performing neurorobotic tactile sensing systems. Furthermore, because the neuromorphic representations are based on biological processing, this work can be used in the future as the basis for naturalistic sensory feedback for upper limb amputees.