I've done covers in a context where people have expected an accurate representation of the songs they know and love, in which case I'd stick to what's on the record, or as close as no one would know the difference, within reason.
But I've also done covers in otherwise original bands, that we've included a cover very much "in our own style" in which case it's no holds barred. Very often radically different covers are great, and can be most inventive.