Are there bidirectional neurons? Neurons are mostly unidirectional, i.e. electrical impulses enter from one end and leave through the other. Are there bidirectional neurons as well, i.e., neurons that have a receiving end and a transmitting end that are not mutually exclusive?
The classical, chemical synapse offers only one-way traffic. Under physiological conditions, the presynaptic neuron releases vesicles at its axon terminal and the postsynaptic neuron expresses the corresponding receptor proteins in the dendritic region. Conversely, the presynaptic neuron lacks the receptor molecules and the postsynaptic neuron lacks the secretory machinery to release neurotransmitter into the synaptic cleft (Purves et al., 2001). 

Note, however, that the presynaptic neuron may express regulatory neurotransmitter receptors that bind its own released neurotransmitter, but these do not generate action potentials. And note that the postsynaptic neuron may release neuromodulators in the synaptic cleft. However, these compounds do not initiate action potentials, but have only a regulatory role.

There is, however, also the electrical synapse, mediated by gap junctions. These gap junctions are protein pores that electrically and chemically connect adjacent neurons directly. In this case, potential changes generated in one cell may freely spread like a wave to other neighboring cells. Note that this mode of electric coupling does not induce action potentials, but it represents the passive spread of voltage differences, called electrotonic coupling (Mylvaganam et al., 2014). Which cell initiates the wave is irrelevant, as the wave can spread to each cell as long as it is connection via gap junctions. Although these gap junctions are referred to as electrical synapses, the terminology presynaptic and postsynaptic applies in a much looser sense, as any neuron in the tissue can act as either one (Sheriar, 2004).  

References
- Mylvaganam et al., Front Physiol (2014); 5: 172
- Purves et al. (eds.), Neuroscience. 2nd ed. Sunderland (MA): Sinauer Associates (2001)
- Sherair BBA (2004); 1662(1–2): 113–37