Can a hyperpolarized neuron fire action potentials? Is there any chance that a neuron could fire when hyperpolarized? In that case, would the spike be different than usual?
Perhaps this is not what you asked, but there's a phenomenon called rebound spiking or postinhibitory spiking where a hyperpolarization causes spiking. This is due to the oscillatory property of membrane dynamics (certain subsets of type-II neurons). Spikes can be evoked after inhibitory current stops. Figure 7.29 from Izhikevich's book:



Here's a partial figure from Hasselmo 2014 (recordings from entorhinal cortex stellate cells).



References:


Hasselmo, M. E. (2014). Neuronal rebound spiking, resonance frequency and theta cycle skipping may contribute to grid cell firing in medial entorhinal cortex. Phil. Trans. R. Soc. B, 369(1635):20120523+.
Izhikevich, E. M. (2007). Dynamical systems in neuroscience : the geometry of excitability and bursting. Computational neuroscience. MIT Press.