$$F_X(x)=\begin{cases} \quad\dfrac{\alpha}{\alpha+\theta}\left(\dfrac x\omega \right)^\theta &\text{ if } x<\omega \\ \\ 1-\dfrac{\theta}{\alpha+\theta}\left(\dfrac\omega x\right)^{\alpha} &\text{ if } x>\omega \end{cases}\quad\text{ where } 0\le x< +\infty.$$
I derived it by having $F_{X|Y}(x|y)=\left(\dfrac{x}{y}\right)^\theta$, which is $Y$ with a multiplicative "shock/noise" given by a $\mathrm{Beta}(\theta,1)$ (max of $\theta$ uniform rvs.) and $Y$ follows a Pareto distribution $F_Y(y)=1-\left(\dfrac{\omega}{y}\right)^\alpha$.
I was not able to find a classification for it. Does it have a name/family?
The distribution is called "split simple Pareto". It is used in actuarial research. It can be obtained as the limit of a transformed beta distribution, see page 184 of this book for additional details:
http://www.amazon.com/Enterprise-Analysis-Lilability-Insurance-Companies/dp/0615133568/ and also this paper: http://www.casact.org/pubs/forum/03wforum/03wf629c.pdf
Although it does not list "split simple Pareto", the following guide may be useful as a quick reference to univariate distributions:
http://threeplusone.com/gec/news/tech-note-survey-of-simple-continuous-univariate-probability-distributions