The authors generalize the Sturmian theory to second-order impulsive differential equations

$(*)$ ${x}^{\text{'}\text{'}}\left(t\right)+p\left(t\right)x\left(t\right)=0$,

$t\ne {\tau}_{k}$,

${\Delta}x\left({\tau}_{k}\right)=0$,

${\Delta}{x}^{\text{'}}\left({\tau}_{k}\right)+{p}_{k}x\left({\tau}_{k}\right)=0$. 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.