The authors generalize the Sturmian theory to second-order impulsive differential equations
. Particularly, a comparison theorem, oscillation and non-oscillation theorems as well as a zero-separation theorem are proved. (Note that all solutions of an impulsive system, given in the special form
, are continuous.) Using comparison results and considering various simple (e.g., periodic, with constant coefficients) test systems, the authors present various sufficient conditions for oscillation and non-oscillation in
. On the other hand, this theory is also used here in the inverse order, to construct impulsive systems of the form
with previously known oscillatory properties. The last section of the paper contains some applications of the main results to nonlinear impulsive differential equations.