I was just browsing through some images of the Bayeux Tapestry, and noticed that for some reason a large number of the subjects are holding their dane axes in what I would consider an 'inverted' grip. Their right hand rests beneath their left on the shaft, which seems counter intuitive to me.

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And then not the best example, but I typically hold a long shafted tool (Rake, shovel, etc) with my dominant hand when resting, similar in style and position to the axe here:
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And then some other sources showing the same grip.

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I did some quick searches, and couldn't find any references to this anywhere. It seems unlikely to just be showing left handed people because it's disproportionately common compared to how common being left handed is, and isn't mirrored in people holding a sword left handed in the same sources. Are there any thoughts or theories on why they're depicted as they are, or possibly anyone well versed in axe combat that has any reasoning, or perhaps has tried it out in reenactment and found any benefit to it?

In reenactment, long hafted, two-handed weapons (spears, axes, glaives etc.) are usually carried like this. The forward left hand is used to hold the weapon steady while the right hand does most of the work (steering, thrusting and so forth).
Dane-axes, glaives and pollaxes also get an advantage against most shield users with this set up as they target the defensively weaker sword-arm of their opponent.

Merry Christmas.

Your Bayeux examples all face the right. Are there men with axes facing the left? If all/most right-facing axemen have left-hand first, and all/most left-facing axemen have right-hand first, then it's artistic convention (not necessarily shared by other works). I've seen this kind of thing with archers in art - those facing one way are drawn left-handed, and those facing the other are drawn right-handed.

That said, one should be able to use polearms ambidextrously. Talhoffer shows both left- and right-hand forward grips, naginata is used ambidextrously in both koryu and modern competition. Sometimes the "off-hand" grip is better, keeping the haft clear of secondary weapons being worn on the left. This is more important for long polearms, but perhaps it matters for shorter polearms. "Off-hand" axe looks good for coming around a shield into the front of the body (i.e., to the left of the shield, from the attacker's viewpoint).

As Baard mentioned, inverted grip is useful when (reenactment) fighting with 2 handed axes and other 2 hand cutting / stabbing weapon. The axe is used less in a wood chopping manner and more as a spear. In a line, I would think the opportunity to land a 2 handed overhead chop is quite rare.

To clarify, using a 2-handed axe "left-handed" means you swing at the opponent's unshielded side. That forces him to turn if he wants to block. Since you don't *have* a shield, that keeps you safer from his weapon, at least briefly!

Matthew

For cutting around the shield to the open side, left hand forward. For thrusting around the shield into the open side, right hand forward, for the angle coming in from the attacker's left side.

I agree with Timo's assertion that you should be able to use a polearm either right or left handed; it's not as awkward as it sounds. Think on the fact that most right-handers tend to put the right hand to the butt of a spear, but the left hand to the butt of an axe. If you do this and can stab with your axe (or even just smack your opponent with the top of it if it doesnt have a long horn) or cut with your spear, you can pretty much already wield a polearm ambidextrously and practice will make perfect!

That all makes tons of sense, particularly the idea of using it to get around a shield, just something I'd never thought of! Thanks as per usual guys, whenever I have a question there seems to be an answer on this forum.

In KDF the staff is considered to be the basis of all other polearms and it is often wielded with the non-dominant hand in the lead. While we don't have a fechtbuch from such an early era I have noticed a lot of recognizable technique and principle in 10th/11th c. artwork and I'm pretty much convinced that the Dane axe and sword and round shield were the original weapons set that proto-KDF was developed on. MS 3227a suggests that the art it describes was hundreds of years old as of the late 14th c.

The other interesting comparison (although much newer) is Jogo do Pau, a traditional portugese method of fighting with large sticks that's very closely linked to longsword arts. They also use a basic grip with the left hand forward on the stick.

JdP has much more in common with historical staff fighting systems than historical sword systems. That's still good, staff fighting is important and there is some overlap with sword arts. JdP has been embraced by some modern longsword practitioners because on the surface it looks a lot like what they want to find, a two-handed grip, and it's easy to run with because most modern longsword competitions are necessarily blunt weapon fights inadvertently governed by blunt weapon rules.

Another way to put it is staff and two handed sword systems are related but single handed sword and two handed sword are more closely related. The difference isn't readily apparent unless you work with sharps a lot and understand the differences between edged weapon combat and blunt weapon combat.