Handedness is the most common and most understandable term. But I'm going for my pedant merit badge, so I'm going to draw upon and explain the other suggestions.
A history and evolution of meaning: Chiral
@DanBron suggests the technical term Chirality:
The word chirality is derived from the Greek, χειρ (kheir), "hand", a familiar chiral object.
An object or a system is chiral if it is distinguishable from its mirror image; that is, it cannot be superposed onto it.
There has been a great degree of speculation as to whether chirality is applicable to the specific chiral orientation or if it is merely the property of being non-superposable. To clear it up, consider the following definitions.
chiral
adjective, Chemistry
1. (of a molecule) not superimposable on its mirror image.
chirality. (n.d.). Collins English Dictionary - Complete & Unabridged 10th Edition. Retrieved April 28, 2015, from Dictionary.com website: http://dictionary.reference.com/browse/chirality
Chiral is the adjectival property of whether an object possesses symmetry, dissymmetry; whether it is superimposable or it is achiral/amphichiral. These have the same definition: not chiral; lacking any chirality in the same way that asexual means non-gendered; neither male nor female. In chemistry, pairs of molecules with opposite chirality are called enantiomer's, sometimes optical isomers; in mathematics enantiomorphs is more common.
chirality
noun
1. the configuration or handedness (left or right) of an asymmetric, optically active chemical compound. Also called dissymmetry
chirality. (n.d.). Collins English Dictionary - Complete & Unabridged 10th Edition. Retrieved April 28, 2015, from Dictionary.com website: http://dictionary.reference.com/browse/chirality
Chiral is a quite new addition to the English language, having only been though up by Lord Kelvin in 1893. The definitions for the word in technical fields and common language are evolving, but the suffix -ity is winning out for the meaning of chirality.
Used to form a noun from an adjective; especially, to form the noun referring to the state, property, or quality of conforming to the adjective's description.
Handedness vs. Chiral
Handedness is commonly used to distinguish between the distinct items of a chiral pair – indeed, almost every definition or explanation of chirality I've heard (or Google suggested just now) uses the mirror symmetry of human hands as a prime example. For instance in physics, for subatomic particles that have a spin, one will be referred to as having left-handed spin, the other as having right-handed spin. It's also applied to describe helix structures (DNA/RNA and threaded screws), electromagnetic and other vector fields where the right-hand-rule is well known, knots and several other geometric and mathematical notions.
JasonC suggests that handedness is a behavioral preference and therefore chirality is not an applicable synonym. However, manual dexterity is as least much a matter of skill as preference. Becoming a switch hitter in baseball requires a great deal of work. Many highly trained and motivated professional athletes do not succeed at overcoming this behavioral preference even though it provides a noticeable advantage to their career. People tend to see their hand dominance (or preference) as a part of themselves, a defining characteristic. In that sense, handedness is as much a physical configuration as a behavioral one.
Chirality is technically correct when applied to people's hands, but almost never used. (That's part of why I like it; it's the best kind of correct.) However, in American English, it's not the right way to ask a person about themselves, mostly because it doesn't communicate the question as several commenters have pointed out. Communication will fail at a rate inversely proportional to the renown of the words (and grammar) used; the words chiral and chirality are not well known by many people without specialized knowledge of science, technology, engineering or math, so don't use chirality. It is not used and will not be understood in common language. If you're pitching a design for a nightlight (or even a horror movie), you can't just say "Are you scotophobic, nyctophobic, lygophobic or achluophobic? Then you'll love this!"
I'm basing my suggestion against chirality and the rest of this answer on the assumption that anyone talking about their designs is seeking to communicate with a broad audience. In short, you question sounds like you're trying to market your designs and marketing messages that are not understood often achieve the opposite of the desired effect. You will drive your clientele away if they don't understand you with minimal effort. If you are specifically trying to be coy and pique their interest by challenging their assumptions, I still suggest you stay away from uncommon technical terms. Unless you are speaking directly and exclusively to an audience you have every reason to expect will know the term chirality, I suggest you don't use it.
Furthermore, chirality might be assumed by people who know the greek root to referred to hand preference. Or a listener might assume you're asking about their hand preference because our left and right hands are the most ubiquitous identifiers of our asymmetry. But technically speaking, chirality could refer to any aspect of human asymmetry, of which there are many: wrists, elbows, shoulders, legs, feet, kidneys, ears etc. etc. all cannot be superimposed on their mirrored pair in the body.
- I lost a foot to a tractor accident.
- My left nostril is slightly larger.
- Every body part I have two of.
Are all correct and applicable answers to the question "What is your chirality?"
But even replacing it with the strict synonym 'handedness' can be confusing.
Eliciting a left-or-right response
I think your first example is best filled by using two words (and to hell with the single-word-request tag). I think it's far more common and understandable to ask:
What's your dominant hand?
This makes it most clear that you're interested in an either-or, left-or-right answer. As some commenters point out, handedness is a spectrum; some people may write with one hand and eat, brush their teeth, cut their food etc. with the other. Nonetheless, when asked directly about which hand is dominant, most people will correctly identify that it's left or right, map it to whatever activities they consider most important, and answer 'left' or 'right'.
I'm basing this on the asker's mention of "state of being left- or right-handed" indicating the desire for an either-or, binary answer. If you want a nuanced answer on a spectrum, use the more open ended "handedness" option.
Inclusive instead of Exclusive language
Similarly, I think it would be very weird to hear:
Our designs do not bias against gender. Our designs are a-gendered/gender neutral.
Our designs do not bias against handedness. (Or, for that matter...)
Our designs do not bias against chirality. Our designs are achiral/chiral neutral.
The distinction itself - gender or handedness - isn't what you'd be biased towards or against. That phrasing implies your designs are for suitable for both gendered and asexual people (or handed and ambidextrous people). I think what you're trying to convey is that your designs are suitable for all people, whether they left-handed, right-handed or ambidextrous. Therefore, I humbly suggest (or rather second Chris H.'s suggestion):
Our designs are ambidextrous.
The prefix ambi- is inclusive, meaning both, unlike a prefix of un- or a- which would mean neither. Saying a person is ambidextrous means they use both hands equally well. Saying an inanimate object or design is ambidextrous implies that people can use either hand to hold, use or operate it equally well. A steering wheel is ambidextrous - it would be hard to imagine one that wasn't (because circles lack chirality). A gear shifter is typically right-handed in cars with a left-handed driver seat (and vice-versa in Britain, e.g.).
And if you're talking about an animate object, ambidextrous is the only way to go!
Our robot designs are ambidextrous.
As you correctly point out, however, you can't remove the prefix from ambidextrous and make any kind of sense. Dexterity by itself doesn't describe handedness or chirality. If you ask someone about their dexterity, you'd get a wide variety of interpretations - my grandmother would start talking about knitting and I'd roll three six-sided dice.
Sliding back into the marketing angle, promoting your product or design as not biased is taking a defensive tone. It is building a subtle connection in the listeners mind between your product and bias. I suggest you try very hard to find a positive or neutral way to describe your designs.
A washbasin with the tap on the center is, then, handedness-blind or chirality-blind, if you will.
It's also precisely ambidextrous.
ambidextrous
adjective
1.1 (Of an implement) designed to be used by left-handed and right-handed people with equal ease:
ambidextrous. (n.d.). In Oxford Dictionaries. Retrieved April 29, 2015, from http://oxforddictionaries.com/definition/english/ambidextrous