I've seen statements of this sort used to motivate the introduction of differential forms, and I'm not sure exactly what's meant.
Obviously if you start by defining differentiation as an operation that takes $n$-cycles to $(n+1)$-cycles, then you can't integrate a 0-cycle to get a (-1)-cycle, but that's silly.
More reasonably, I guess there's an issue with orientation, but Riemann surfaces are orientable so we could define integrals up to orientation which sounds like it would be "good enough."
Is that all that's being said here? Or is there some deeper issue?