What does the time-reversibility of Verlet (or other) integration mean? What does the time-reversibility of Verlet or any other integration method mean? The wikipedia article about it is very complex, unclear and confusing.
And how can I determine whether a method is time reversible or not?
For example the classical Störmer-Verlet method
$ x_{n+1} = 2x_n - x_{n-1} + a(x_n) · dt^2 $
How is this time-reversible? If I change the sign of the timestep dt, because of the square nothing changes.
 A: Yes, it is exactly that.
The time reverse of the explicit Euler method is the implicit Euler method. $y_{n+1}=y_n+f(y_n)\,dt$ gets reversed to $y_{n-1}=y_n+f(y_n)\,(-dt)$ and after index shift $y_{n+1}=y_n+f(y_{n+1})\,dt$.
The same for the symplectic Euler methods. 
\begin{array}{lll}
forward:&x_{n+1}=x_n+v_n\,dt,& v_{n+1}=v_n+a(x_{n+1})\\
reverse:&x_{n-1}=x_n-v_n\,dt,& v_{n-1}=v_n+a(x_{n-1})\\
shifted:&v_{n+1}=v_n+a(x_n)\,dt,& x_{n+1}=x_n+v_{n+1}\,dt
\end{array}
(Velocity) Verlet is a combination of an explicit and implicit symplectic Euler step, thus invariant under time reversal.
\begin{align}
v_{n+1/2}&=v_n+a(x_{n})\,dt/2\\
x_{n+1/2}&=x_n+v_{n+1/2}\,dt/2\\
x_n&=x_{n+1/2}+v_{n+1/2}\,dt/2\\
v_n&=v_{n+1/2}+a(x_{n+1})\,dt/2
\end{align}
Elimination of $x_{n+1/2}$ gives velocity Verlet, elimination of the integer-indexed velocities gives the Leapfrog method, elimination of all velocities gives the basic Stoermer-Verlet method.
The first reversion-invariant Runge-Kutta methods are the (necessarily implicit) trapezoidal and midpoint methods.
\begin{align}
&trapezoid:&y_{n+1}&=y_n+\tfrac12(f(y_n)+f(y_{n+1}))\,dt\\
&midpoint:& y_{n+1}&=y_n+f(\tfrac12(y_n)+y_{n+1}))\,dt
\end{align}

The advantage of reversion-invariant methods is that the error in scalar invariants is also reversion invariant and thus an even function. For the second-order Verlet method one would expect from the order alone that the first term of the local error in energy and momentum is $O(dt^3)$, but since the odd powers are missing the actual error is $O(dt^4)$.

Of course, the same or better can be achieved with the classical RK4, but symplectic Euler and Verlet are better suited for real-time simulations and provide, because of "symplectic", global (very slowly eroding) error bounds on conserved quantities of the dynamical system.
