We discuss properties of distributions that are multivariate totally positive of order two (MTP$\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}$) related to conditional independence. In particular, we show that any independence model generated by an MTP$\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}$ distribution is a compositional semigraphoid which is upward-stable and singleton-transitive. In addition, we prove that any MTP$\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}$ distribution satisfying an appropriate support condition is faithful to its concentration graph. Finally, we analyze factorization properties of MTP$\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}$ distributions and discuss ways of constructing MTP$\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}$ distributions; in particular we give conditions on the log-linear parameters of a discrete distribution which ensure MTP$\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}$ and characterize conditional Gaussian distributions which satisfy MTP$\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}$.