How do neurons inverse an action potential?
The University of Texas site where you obtained this picture states in section 6.7:


  What about the transmitter substance that is released by the inhibitory interneuron in the spinal cord? The transmitter substance is glycine, an amino acid which is used frequently in the central nervous system as a transmitter that produces inhibitory actions. It is not the most common, however. The most common transmitter with inhibitory actions is gamma amino butyric acid (GABA).


Glycine receptors activate a chloride-selective transmembrane channel, which is predominantly expressed in the spinal cord and brain stem (Rajendra et al, 1997). Chloride hyperpolarizes the neuron, thereby inhibiting the motorneuron controlling the flexor muscle, and thereby facilitating the extensor muscle to contract. If the opposing flexor muscle would not be inhibited, the extensor muscle would be counteracted and the knee jerk reflex would be less efficient, if present at all.

Typically, an inhibitory interneuron will only release inhibitory neurotransmitter. In the example here, the opposite circuit to bend the knee would activate the flexor muscle. This would be accompanied by opposite events. However, the inhibitory F neuron above in your figure would not reverse its action though, it will simply be silent, thereby allowing the acetylcholine from the E neuron do its job. In more general terms, neurons cannot reverse their action from inhibitory to excitatory. Their outputs are typically simply the release of a pre-defined vesicle pool, which is a yes/no event without further qualities. In other words, an inhibitory neuron will be inhibitory and it cannot change the nature of the neurotransmitters released (excitatory versus inhibitory) based on different inputs.   

Reference
- Rajendra et al, Pharmacology &amp; Therapeutics (1997), 73(2): 121–46