The Shulchan Aruch (27:11) writes that a person wearing Tefillin must ensure that specifically the black side of the straps faces outward (listen to audio for precise citation). And the Gemara in Masechet Mo'ed Katan (25) records that Rav Huna's straps were once inverted, such that the black side faced inward, and he observed forty fasts to atone for this violation. This indicates that the Sages viewed this requirement with particular severity.
It should be noted that this Halacha applies only to the part of the straps with which the Tefillin Shel Rosh is tied to one's head, and to the part with which one ties the Tefillin Shel Yad to his upper arm muscle. It does not apply to the straps that hang down from the Tefillin Shel Rosh, or to the part of the straps of the Tefillin Shel Yad which one wraps around his forearm.
One who notices after removing his Tefillin that the strap was inverted in one of those places should preferably observe a fast day. If he finds this difficult, he should at least give to charity the amount he spends on a day's meals. The requirement to fast or give charity applies only if one removed his Tefillin and then realized that the strap was inverted the entire time. If one fixed the straps while he was wearing them, such that he did wear them properly for a period of time, there is no need to fast or give charity.
Given the severity with which Halacha treats this matter, one must ensure when putting on his Tefillin to check either with a mirror or with his hands that the black part of the strap faces outward in the areas mentioned. Quite often when one ties the Tefillin Shel Yad around his arm the strap becomes inverted, and thus extra caution is required in this regard.
Summary: One must ensure that specifically the black side faces outward in the part of the strap of the Tefillin Shel Rosh that goes around his head, and the part of the strap of the Tefillin Shel Yad with which one ties the Tefillin around his upper arm muscle. If one realized after removing his Tefillin that the strap had been inverted throughout the entire time he wore them, he should observe a fast day or at least donate to charity the money he spends on a day's meals.
