$PV$ work for irreversible expansion vs. irreversible compression?

Assuming we have an ideal gas expanding against a pistom in a cylinder isothermally, then the pressure drops due to viscous strain within the gas and we can extract less work than for an reversible expansion. But if we want to compress the gas irreversibel, back to the initial state the pressure must be higher than for the expansion to overcome the viscous strain. What I do not understand is that the integrated PV work is higher for.the compression than for the expansion though we have as far as I know the same work loss as friction and the same Volume change. Could you please help me?-

The work done by the gas on the piston during irreversible expansion is less than the work done by the piston on the gas during irreversible compression. In either expansion or compression, a rough approximation to the force F on the piston in the irreversible behavior of the gas is given by: $$\frac{F}{A}=\frac{nRT}{V}-\frac{\mu}{V}\frac{dV}{dt}$$where the first term represents the equilibrium equation of state behavior of the gas, and the second term represents the viscous damping. The parameter $$\mu$$ is proportional either to the actual viscosity or to the turbulent viscosity (depending on whether the flow is turbulent). Note that, in expansion, the viscous term makes a negative contribution to the force on the piston, while, in compression, the viscous term makes a positive contribution to the force on the piston.