I am interested in knowing whether there is any discrete probability distribution similar to Poisson but also extended in the negative value part (i.e., it can take negative values and there is no a lower limit for the values it can take).
Is there any family of discrete distributions meeting these requirements?
More requirements: The shape is not necessarily symmetrical. There is one more condition to be met: values close to $-1$ and $1$ are very likely to be observed, but value $0$ is very improbable. Besides this, the probabilities of observing values greater than $1$ or lower than $-1$ monotonically decrease (more or less like a Poisson distribution does, but also in the negative side).
Answering the comments below: yes, my data follow a sort of bimodal distribution with modes in $1$ and $-1$.