Symmetric sigmoidal function with asymptotes?

Question

Consider the cubic function:

f(x) = x + ax^3

Simple cubic function

This function has several of the features that I need, which are:

1. crosses the origin (0,0)
2. If x > 0 then y <= x
3. If x < 0 then y >= x
4. Symmetric: f(x) = -f(-x)
5. parameter "a" regulates the departure from the y=x line

HOWEVER, instead of "bending up/downwards" like the cubic function, in addition to all properties listed above, I would like a horizontal asymptotic function that gradually curves and "levels off" towards a parameter value. So, the function should have just one root (0,0).

It feels like a function from the family of sigmoid curves could do the trick, but after trying for many days I still have not found the right equation.