Nowhere in the Bible is there anything which even hints that men must cover their heads. Nowhere. Not even tangentially. As far as I know, outside of war gear, there is no mention in the Old Testament of anyone wearing anything on their heads. I'll get to t'fillin later. The only reason I know of which explains why men cover their heads, especially during religious services, it Christians have a tradition of removing their hats when they enter a building. Also for no apparent reason. It's meant as either an insult or minor rebellion.
Tsistit - fringes. The Bible clearly states that men must wear garments which have a fringe with blue. The WTF here is that while orthodox men will wear a chemise with specially knotted fringes under their shirts** to meet this requirement, the fringes are white, with black stripes on the material above the fringe. I wonder what part of "blue" they have failed to understand for 1500 years. Numbers 15:38 - "Speak to the children of Israel, and tell them that they should make themselves fringes in the borders of their garments throughout their generations, and that they put on the fringe of each border a cord of blue". A minor WTF is the order is given to all Israelites, not just the men, but women have never followed the command, or at least haven't in the last 100 years or so.
T'fillin is a set of two little black boxes with long leather straps attached. Every morning orthodox Jewish men wrap the straps of one box on the hand, in a winding across the forearm which is meant to evoke the Hebrew letter shin. The other box is strapped like a headband, with the box centered on the forehead. In the box is the Bible passage which a very literal person might concede commands this rather bizarre activity. Once again, a Biblical admonition is taken to a logical conclusion well past its intent, pretty much destroying the beauty of the scriptural passage.
The passage (Deuteronomy 6:8) says:
And thou shalt bind them for a sign upon thine hand, and they shall be as frontlets between thine eyes.
The lead-in to that line is important:
Hear, O Israel: The LORD is God, The LORD is One 5 And thou shalt love the LORD thy God with all thine heart, and with all thy soul, and with all thy might. 6 And these words, which I command thee this day, shall be in thine heart: 7 And thou shalt teach them diligently unto thy children, and shalt talk of them when thou sittest in thine house, and when thou walkest by the way, and when thou liest down, and when thou risest up. 8 And thou shalt bind them for a sign upon thine hand, and they shall be as frontlets between thine eyes. 9 And thou shalt write them upon the doorposts of thy house, and on thy gates.
The message is to remember that there's only one God, and you love Him. It's not saying to literally strap something to your hand or forehead, or tack it to the door post (but they do - every door frame in a Jewish home is supposed to have a mezzuzah - a little box with this prayer rolled up inside it- tacked on it). IMHO by literally doing what 8-9 says, one destroys the poetry of the command.
**The knots are not called for in the Bible, they are some medieval rabbi's guess at quasi-kabalistic numerology. The knots have some mystic numerical significance.