Echar rostro refers to the situation when one is hanging out with the intent to be seen by others as cool and collected. For example, when you are dando el roll (taking a stroll) or you are riding your convertible with sunglasses, you are said to be echando el rostro.
The idea of the expression is very graphic, because in Spanish it literally means to give away, (share) your face with others.
One variation of this expression is "echar el guante" (throw the glove) and think here of an image of a white-gloved lady taking a stroll in the park.
Toño: qué pez Rulas, vi unas reinas en los portales, vamos a echar rostro?
Rulas: bambi, tu di rana y yo salto.
Toño: hello Rulas, I saw some cute girls in the shopping mall, shall we go and hang out there?
Rulas: of course, you take the lead.
Check the picture bellow, with Mauricio Garces (champion of galant movies of Mexico) is echando rostro.