I think you’re right about shape, but I thought the overall orientation/direction was as I said - I’m very happy to be corrected if I’m wrong.

The reason I relate it as such is because a comet’s tail doesn’t trail parallel to its trajectory, but rather parallel with the sun’s rays. As the heliotail’s orientation and shape are determined by the Interstellar Magnetic Field’s direction and strength

Thank you, good distinction.

there is a heliotail, which is a comet-like tail that is inflated by the solar wind of particles streaming off the sun, but it's more of a tail of the entire solar system than just streaming behind the sun like you might be thinking.