"The sun seems to linger for several minutes in its passage across the sky, before beginning to double back ... at dawn for two or three days ... on the first days of summer and winter", reports Richard Cohen in the December 19, 2010 New York Times editorial "There Goes the Sun", which is reproduced in part above. Cohen gets the etymology right, but the phenomenon wrong: the Sun cannot appear to "double back" at dawn, otherwise it would set!
So what exactly stands still on the solstices? It is not the sun itself, which continues its apparent motion through the sky as always. It is the sun's declination, its celestial latitude. The Sun's altitude, which is the complement of the declination of the Sun at solar noon minus the latitudei.e., the latitude of the Earth position for which the Sun's altitude is being calculated, changes very little near the solstices. Suppose we make a movie of the Sun at solar noon at a given location with frames taken each day near the solstices and near the equinoxes. In the solstice movie, the sun would remain at almost at the same altitude; in the equinox movie, the Sun's altitude would change visibly.
Check out these tables of the declination of the sun: near the solstices, the Sun's declination changes less than an arcminute per day; near the equinoxes, the sun's altitude changes over twenty arcminutes per day!