I think it's more that the left believes government has a responsibility to guarantee equality of opportunity, not equality of outcome in what is assumed to be a fundamentally inequitable system. The big philosophical difference to me is in what each side believes the proper role of government should be. The right (tending to be the side that supports the military, police, harsh criminal penalties, etc.) believes government's only legitimate purpose is enforcing the laws and defending the borders, in other words the exercise of violence, whereas the left also believes the government has a duty to provide for the social welfare of its citizens in exchange for the power granted to it by social contract.
And when the right says it wants liberty, what it tends to mean is that it wants liberty from the social obligations that the left tend to want government to support.