After planking, owling and leisure dive, now there is the trend which is called "Batmanning". Obviously it is a made-up word. Is there a proper substitute for it? What is the verb that defines the act when a bat hangs from somewhere?


Question: How to reduce vehicle swaying (body sway)
Answer: You are very close to answering your own question:


  ... what can i do to reduce the swaying?


What you're looking for is an upgraded sway bar.  This is "... usually a torsion spring that resists body roll motions."  This is in contrast to a strut bar "... designed to reduce ... strut tower flex by tying two parallel strut towers together."

In general, if your goal is to reduce body roll, one of the cheapest ways to do so is to upgrade the sway bar(s).  Strut tower bars will have a much more limited effect and should usually be considered after upgrading tires, sway bars, struts and springs (i.e., much later).

Caveat: increasing the roll stiffness will change your under/oversteering biases.  Specifically, a stiffer bar in the front generally increases understeer.  Likewise, a stiffer rear will increase oversteer.  That said, I had a 97 Integra some time back and installed an adjustable rear sway bar.  I adored its steering responsiveness in the summer but also managed to scare myself several times in the winter....