“You possess the amazing ability to repair any item using a roughly similar item. Fix a trail carbine with a hunting rifle, a Plasma Defender with a laser pistol, or even power armor with metal armor. How does it work? Nobody knows... except you.”— In-game description
There are 13 categories of weapons that the perk will enable repairing within: unarmed, one-handed melee, two-handed melee, energy pistols, energy rifles (laser), energy rifles (other), energy weapons (heavy), one-handed guns, two-handed automatic guns, bolt/lever-action guns, miniguns, grenade launchers and missile launchers. Likewise, clothing can be used to repair each other within their category (light, medium, heavy). Finally, almost all headgear can be repaired with any other headgear.
Weapons are categorized, to some extent, by the method in which they are fired. For example, the bolt-action anti-materiel rifle can not be repaired by a sniper rifle because the sniper rifle is a semi-automatic firearm, but it can be repaired by the hunting rifle as it is also a bolt-action weapon. However, some discrepancies arise with other weapons. For example, the 12.7mm SMG is categorized as a two-handed gun and therefore can not be repaired by the 9mm and 10mm SMGs, since they are one-handed guns. A mostly complete list of all weapons that fall into specific categories can be found below.
A simple way to tell what weapons can repair each other is to look at the way that the player holds the weapon, in addition to the general type of the weapon (Guns, Energy, Explosives, etc.). For example, as the player holds the plasma rifle in the same way that they hold the laser RCW, Jury Rigging lets the player repair them with each other, even though one is laser and one is plasma. However, since the laser rifle is held differently (more like a bolt/lever action gun), one cannot use it to repair the laser RCW, nor vice versa.
This perk reduces the need to effect repairs using weapon repair kits or non-player characters. It also allows the player's character to reliably repair their weapons with similar, much cheaper ones from fallen foes after or during combat without having to seek out an exact match. Characters who use low-durability weapons and picked the Built to Destroy trait, or find themselves using surplus or over/max-charged ammunition often, reap greater benefits from the perk. It can also be a big help to those who favor certain unique weapons like the YCS/186 or Oh, Baby!, whose normal repair counterparts are rare or expensive, or weapons that do not have a standard repair counterpart (like the X-2 antenna, which is a completely unique weapon). This goes for armor, as well, meaning that even Remnants power armor can be maintained on the cheap with a set of metal armor.
Note: Jury Rigging also allows the player to repair common items with rare or powerful items, potentially causing the player to lose these items accidentally, so it's advisable to pay close attention when repairing items.
The riot gear helmet and its variants are the only headgear classified as "Heavy." Since they can already repair one another freely, Jury Rigging only has the effect to allow them to be repaired by heavy armor. The other helmets and masks from Lonesome Road are classified as "light" and can be Jury Rigged with any other headgear.
All marked men armor regardless of having light or medium weight can be repaired with leather armor or raider armor, which is considered light weight.
There are certain quirks in item category that the Courier can use certain specific subtype of items to fix the other, but not vice versa. Example should be a combat knife used to fix a bowie knife, or merc grunt outfit used to fix naughty nightwear.
Via quite extensive GECK testing the variables involved are almost certainly skill (skill drop down directly below Clip Rounds and Regen Rate) and Type (Animation Type on the Art & Sound Tab.) This taken into account, the grenade machinegun is excluded due to being Explosives/TwoHandHandle where the missile launcher et. al. are Explosives/TwoHandLauncher, the Tesla cannon alone in Energy Weapons/TwoHandLauncher while the plasma caster and Gat. Laser are TwoHandHandle, the holorifle perhaps mis-classed as Energy Weapons/TwoHandRifle rather than TwoHandRifleEnergy, and the shoulder mounted machine gun Guns/TwoHandLauncher rather than TwoHandHandle like other minigun types. Given the animations for these weapons (for example the grenade MG (Expl) is basically a re-skinned incinerator (E Weaps) and the Tesla cannon (Energy Weapons) is shoulder fired more like a missile launcher (Expl)) the exclusions were most likely an unfortunate side effect of coding that makes the perk drastically more portable for add-ons and mods. The binoculars were most likely intentionally excluded due to not having realistically usable repair parts for any weapon category.