back to the basics

Newton’s equation of motion \(m\ddot{x}^i = -\partial^i\Phi\) has two interesting features: It allows for time reversal due to the second derivatives and exhibits inertial motion at constant speed in the force-free case. Can you have one without the other? And for bonus points: in what way are these two properties incorporated in the Schrödinger-equation as a first-order differential equation?