In a survey about the trek of grey whales along the west coast of the United States an observation was made of a flukeless individual that nevertheless kept up with the pod. That sounds astonishing, except that this can be explained.
The peduncle of whales and dolphins viewed from above (or underneath) narrows, so it finds little resistance moving up and down and the fluke can be moved more forceful. The bundles of muscle are vertically flattened. This implies that by moving its peduncle sideways a dolphin also can exert thrust. Said grey whale has probably advanced like that. By bending the peduncle to one side also an incline in that direction can be made as the body is slowed down at that side. A dolphin primarily navigates with her peduncle. The pectoral fins serve the fine-tuning.
Some while ago I have imagined a wetsuit in which my legs were not aside each other, but on top of each other, the toes of one foot upon the heel of the other, so sideways I would have a broader push-off and up and down less resistance. Unfortunately we are not built to move like that and therefor essentially swimming like a dolphin, combining up and sideways together with rolling on the length axis is not in our locomotional blueprint.
The closest to that comes like children playing 'plane flying', with the arms as wings sideways and when you moved them askew you 'made a bend'. Now water is over 800 times denser than air so you don't need much speed to 'fly' through the water. Moreover, under water you are virtually weightless. Aces in short track underwater manoeuvring are seals and particularly sea lions, while in the heavy weight camp humpbacks, whose pectoral fins measure a third of their body length, are the most lively.
Still we can lick a taste of how it feels to move through water in a sideway fashion. Simply by laying on your side. I do that with monofin and waterwing vertical and lengthwise parallel to my body for best result. And that is noticed, by Dusty. She finds that rather interesting, but I do believe it's more because it is exceptional. When you want her attention it works best to do something outside 'normal' swimming behaviour. Because all those people who only try to touch her, they are a dime a dolphin, eh, dozen.
Fortunately us humans have the gift of imagination. In the water I may be a artistic reflection of a dolphinman, but in my dreams I glide in celestial undulation between the sparkles the sun casts in a crystal clear ocean.