Stare at awkward long enough and I think you'll agree with me. What an awkward word, with that wkw in the middle. It turns out to be a combination of the Middle English adjective “awk” and the directional suffix “-ward.”
It seems “awkward” was coined in the 1300's in Scotland and northern England, where it meant “turned in the wrong direction”. The word "awk" meant the wrong way round, backhanded. Other possible meanings are sinister, ominous, perverse.
Here’s an example of the sinister/ominous meaning of "auke" from Philemon Holland’s 1600 translation of Livy’s history of Rome and the Roman people. In this passage Livy refers to those who disparage the Roman practice of augury:
Now let them mocke on and scoffe at our religions. Let them deride our ceremonies. What makes matter (say they) if those pullets pecke or eat not? What if they come somewhat late out of their coupe or cage? What if a bird sing auke or crowe crosse and contrarie? How then?
And here's a late example from 1674, where perhaps perverse is meant. It's in a scientific treatise from the 17th century clergyman Nathaniel Fairfax:
What we have hitherto spoken, will seem to have less of auk in it,
That is, what we have hitherto spoken, will seem less perverse. Fairfax was keen to use native English words only, and I suspect that by 1674, having “less of auk in it” already sounded old-fashioned, or dare I say, awkward. (I have more on Nathaniel Fairfax and the context of this quotation in an appendix. It interests me because of a connection to the history of science. He seems to have been exploring some of the thoughts that gave rise to calculus at about the same time.)
For an early instance of "awkward", there's the Middle English poem Pricke of Conscience (1340): the world thai all awkeward sette (they turned the world all awry).
A bit of etymology
“Awk” is Scandinavian in origin. Its equivalent in modern Swedish is “avig”. Suppose you were to put a shirt on back-to-front, this in Swedish would be “att ha skjortan avig”, literally to have the shirt the wrong way. There's a German word "Abweg" meaning the wrong way, which looks as if it ought to be related, but so far as I can tell it isn't.
I can't account for why, but it tickles me that the “ward” in awkward has something to do with direction, as in northward, onward, backward, inward, and so on. We can perhaps think of awkward as equivalent to the non-existent word wrongward.
My Shorter Oxford Dictionary tell me that the suffix ”-ward” gives the meaning of having a specified direction, and is connected with the Latin verb vertere (to turn). I find that an especially fruitful piece of etymology as it helps us to think of “–ward” as having the meaning turned in the direction of. So: turned in the direction of in, turned in the direction of out, turned in the direction of north, etc. Then there's "toward", and the interesting case of "untoward". In Middle English there was a word “fromward”; which in Old English apparently meant "about to depart; doomed to die; with back turned."
“-ward” can in principle be added to any location, to suggest progressing or pointing towards that place. As in she raised her eyes heavenward. Or this sentence from H. G. Wells, The War of the Worlds (1898): In the road that runs from the top of Putney Hill to Wimbledon was a number of poor vestiges of the panic torrent that must have poured Londonward on the Sunday night after the fighting began. And in a recently published legal history of New York we find: It was not until the colony became a state that the pendulum of emigration and settlement swung New Yorkward.
I thank the excellent Grammarphobia blog  for calling my attention to the awkwardness of “awkward”. It puts me in mind of the opposite case, the mellifluousness of “mellifluous”. A curiosity I had something to say on back in May.
“Awkward” and “mellifluous” are autological words, words that describe themselves – or so it seems to me.
 The blog is the source of many of the foregoing quotations and I've even plagiarized the title of this post from it. You'll find more information in an email from the blog editors reproduced in the appendix.