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I was hunting for a unicode version of the implies sign, which I usually write as = plus >, to produce => (implies) and its opposite <= (is implied by) with the mutual <=> that they imply each other. I found then eventually in unicode 21D0 ⇐ and 21D2 ⇒ and 21D4 ⇔ but I also see that they don't align, that ⇐ and ⇒ are not mirror images, even if I look at other fonts. I add this then to the inability to associate p with dots and umlauts — and perhaps head towards giving in to use of LaTex for writing maths.

However, I am distracted by the very many mathematical symbols within unicode, after many years of teaching this very subject. 

Here are those I recognise and used in teaching, in unicode order....

as in 3! the factorial, the arithmetic operators + - * /, ratio :, less than etc < = >, index ^, all the brackets {[()]} and ~ 'twiddles' for has some sort of relationship with'. But I wouldn't use * and ^ in class, only on the keyboard. I didn't use the old division sign, ÷ 00F7. On the keyboard, I use fractions when I know they're available, ¼½¾ and I use all the Greek letters I can find. In further maths I'll add dots to all sorts of letters, so that for example ÿ is the second differential with respect to time, d²y/dt² and—as I've written several times now, these are not all available in unicode. In typing, I want to be able to use almost anything as an index and I have learned that subscript and superscript markers fail, so when I can I use the unicode versions ᵅᵇᵈᵉᵍᵏᵐᵖʳᵗᵘᵛˣʸ ᵅᵝᵞᵟ, which, you notice, are inconsistent in size and elevation, as are the number indices ⁰¹²³⁴⁶⁷⁸⁹. Much the same applies to subscripts, 

 xₐₑᵢₒᵣₓᵤᵥₔᵦᵧᵨᵩ, y ₕₖₗₘₙₚₛₜ. Look at 1Dxx and at 0280-036F. These are called modifier letters. Also 16Fxx, 209x

At last the how-to for p᷉ deduced; used 1DC9, p᷍  and p̂ similarly and perhaps the mandarin tones for pinyin, cha᷄,  shi᷅  see 'combining diacritical marks', 1DCx;  03xx is earlier, for p̂, p̄, ṗ, p̈;  huā, chá, wŏ, shì.  I wonder if I have any uses at all for  a̽, a͓, 2ͭ, perhaps a²ͭ. But I did manage  a³  ͭ, which is a new achievement.

I'd very much like an improvement to ⁰⁴⁵⁶⁸⁹ and ₀₁₂₃₄₅₆₇₈₉;  x¹⁰ is so very bad. Wikipedia on this topic.

Here's an attempt to use superscript, x¹², y ¹⁰, z³⁴ ; that's supposed to be x¹², y¹⁰, z³⁴.

There are more fractions available at unicode 2150 and, just above those the specialist number classes which I'll let you puzzle over: ℂ,ℙ,ℚ,ℝ,ℤ - though I'm not sure I ever used ℙ to indicate primes. The fractions are ⅑ ⅒ ⅓ ⅔ ⅕ ⅖ ⅗ ⅙ ⅚ ⅝ ⅞ and I note that these too are not consistent in display.

Among the arrows, I used ⇐⇔ and, rarely ⇍ ⇏  ↦ but I did not use ⇌ outside chemistry and I did not use ↔ → ← at all. This brings us (at last, I say) to the section actually labelled mathematical operators. Here are the first two lines in the unicode library:

We jump to Unicode 2200: I can identify some but not all of these: ∀ is 'for all', as in this applies to all reals, ∀ℝ. I never used ∁ to indicate a complement (though perhaps I should have). ∂ is a partial differential;  I used 'there exists' ∃ and its opposite ∄; ∅ is the empty set, though in typing I'd use ɸ (phi) where I far prefer it to have a slant in the italic form, 𝜙.  I would on occasion use ᵠ or 𐐐, a curly phi, as if it were a separate symbol, just as one would use a curly w, (ω, lower case omega) to represent cube root of -1. I would replace phi with ø at 00F8 as my preferred character in typing for an empty set, the nothing-not-zero character. I'd use ∈ and ∉ for 'is an element of' in set theory and very much less often the reverse, 'contains as a member' ∋ and ∌ but not the two small versions. I'd have rare occasion at A-level for the compound multiplication sign ∏ but would use ∑ often in the study of series. Obviously I'd use ∕ and √∛∜; ∝ ('is proportional to', but I'd write it much larger) and ∞ (infinity). I'd not use ∓, minus or plus, only the other one ± (shift plus, 00B0,  which goes with 'not equal' ≠ also available directly on my keyboard). I do not recognise the 'not' sign, ¬, where I'd have the line above the relevant letters, and Ā would be read out loud as A-bar or not-A. quite often one would use A' as not-A.

Note, too, that there is a pretty complete set of specifically mathematics alphanumerics starting at 1D400. Examples: 𝐀𝐁𝐂𝐚𝐛𝐜 𝐴𝐵𝐶𝑎𝑏𝑐 𝑨𝑩𝑪𝒂𝒃𝒄 𝒜𝒞𝒟𝒶𝒷𝒸 𝓐𝓑𝓒𝓪𝓫𝓬 𝔄𝔅𝔇𝔞𝔟𝔡 𝔸𝔹𝔻𝕒𝕓𝕕 𝕬𝕭𝕮𝖆𝖇𝖈 𝖠𝖡𝖢𝖺𝖻𝖼 𝗔𝗕𝗖𝗮𝗯𝗰 𝘈𝘉𝘊𝘢𝘣𝘤 𝘼𝘽𝘾𝙖𝙗𝙘 𝙰𝙱𝙲𝚊𝚋𝚌 𝚨𝚩𝚪𝚫𝛂𝛃𝛄𝛅 𝛢𝛣𝛤𝛥𝛼𝛾𝛿 𝜜𝜝𝜞𝜟𝜶𝜷𝜸𝜹 𝝖𝝗𝝘𝝙𝝰𝝱𝝲𝝳 𝞐𝞑𝞒𝞓𝞪𝞫𝞬𝞭   Given the difficulty of switching to a specific unicode, I see more point in offering these as a font choice.

I do not recognise the other characters. ∟, the right angle, would always be written by me with a tiny square in the angle, as ⦜ 299C. In the same way I'd use angle ∠ABC, but with a far larger symbol, more like a capital italicised L, 𝙇ABC, the same size as the letters in use. 𐝆ABC uses 10746 (Linear A) and I think is a good choice.

I recognise very little of the next five lines in unicode, beyond the integral signs. ∴ is therefore and ∵ is because (used very rarely in school). In set theory I'd use ⊂ ⊃ ⊄ ⊅ ⊆ ⊇ ⊈ ⊉, ⋂, ⋃ liberally, but then reduce the symbols in common use so as to help the dyslexics be less confused. The remainder of these here are either near duplicates (e.g. ≤, ) of symbols already identified or unrecognised. I would use ⊗ for the cross product and might very rarely use ⊕ to indicate that this was a different meaning to addition. I don't recognise, for example ⊨ as 'true', or ⊲ as 'normal subgroup of'; I very rarely used the logical and/or ⋀ ⋎ and their complements ⊼ ⊻  (nand and nor)

There are even more maths symbols at 27C0 and 2980. Of these ⟌ (27CC) is the opening of the long division symbol, where one would have a over-arching line above as a class of bracket (one of those things that works in handwriting but not in typing).

I recognise a sequence of this last group as logical symbols (⩑⩓⩕⩗⩚⩜⩞⩢ for example) without knowing whether I have ever or would ever use them.

Non-mathematical symbols

ＡＢＣＤａｂｃｄ is different from ABCDabcd, found at FF20-40. 

Higgs and Hill, the builder, would love to know of 𐋅 the Carian letter II, since it is so very close to their logo. 𐋅 s is even closer. there is also ℍ at 210D

Quite a few letters are available reversed or inverted. Wikipedia has helpful product (found after I'd played for the odd hour).

ƎɘϽↄꟻꟼᴎᴙ     Canadian aboriginal ᖆᖈᖉ  ᖴᖵᖶᖷ ᗷᗺ ᘮᘯᘰᘳ  ᙎᙏᙓᙐ.  In letterlike 2100-214F, I found ⅁⅂Ⅎ℺⅋ⅎ. In Lisu, A4D0-FF, there are ꓕꓘꓞꓤꓨꓩꓭꓯꓱꓵꓶꓷ  

After some hunting, I found a site or two doing the swap with a bit less work than I had been doing, inserted to the right. Using that to spot the difficulties (instead of the very slow way I had been doing it), I'll point to a few solutions and issues. I'll use reversed for a left-right swap rotating on a vertical axis, and inverted for the rotation on a horizontal axis. So N reverses to И and A inverts to ꓯ.

The reversed t appears to be an f, ɟ while the f is ʇ, an inverted t. Uniform and November are used as inverts, u and n. g reverses to a rho, ϱ.

I found:  N in the Russian, Cyrillic I , И, 0418  

a reversed or inverted S A6EF ꛯ, and 01A7 Ƨ, 01A8 ƨ.

The reversed and inverted J proves incomplete: ꓩ A4E9 Lisu as above, but I have not found a lower case j mirrored in either axis. Only close, we have ꌌꋍ though ꒓꒒ A492&3 are quite good. Ꙇ ꙇ A646-7 can blow up to  Ꙇ ꙇ ; Ɩ is 0196, an iota, so ίἰἱ, almost serves, though one would prefer a descender component in the letter.  𑚯 and  𑚮 occur in Takri 116AE&F, plus  𑫗 11AD7  and𞡄 1 E844 which behaves oddly, as if reversed already . See this mixture  𑚯 J  𑚮ꓩJ ᒐ 𑫗 .  ꮣ ABA3 

H has no difficulty but h has. 𑢯𑣏𑣄 are around 118Ax 𑀛 is 1101B; 𐨜 is 10A1C; 𐔼 at 1053C looks good in comparison to the ɥ in the text box; here they are together, ɥ and 𐔼, so hoax would be xɐoɥ or xɐo𐔼; ɥ is 0265. 𐋍 𐋌 at 102Cx; Ⴙ Ⴗ 108x Ⴗ 1087.

letter a:  ɐ is 0250;  ᵄ 1D44, ᵊ 1d4A; ɒ is 0252 and nearby is ʁ, 0281,  ᴚ 1D19 and ᴙ 1D1A, 

and L (there's the gamma, Г) so I can write ƧИIOϽƧ ИAI⅃Uᒐ ꓷIVAꓷ and SNIOϽS NꓯIꓶꓵꓩ ꓷI𖩅ꓯꓷ.

The V was found at Linear A 10620, 𐘠 or (better) Miro 𖩅 16A45. Alternatively, ꛎ at A6CE. 

The A ∀ is 2200, enlarged, or better still,  Lisu A4EF, ꓯ.

As written above, the letters  f and t  might be mirrored with each other, or  𑫚 11ADA  ʇ 0287  [07D4 reverses direction, ߔ]  

Here's a list of inverted codes. https://www.upsidedowntext.com/unicode 

List of reversed characters can be done using U+200F, but use of this character has security implications. I failed to get it to work, too. Try here, or similar. 

It looks to me as if the mirrored text service swaps unicode, which is why the lower case j does not mirror.

Can you point at a character and discover what unicode is is? yes; if you can copy the character, here. So, for example, I identify the characters used in zʎxʍʌnʇsɹbdouɯןʞɾıɥƃɟǝpɔqɐ here           ➤

Similalrly, the unicode for zγxwvuɟƨɿpqonmlʞᒑi⑁ϱʇɘbɔdɒ is on the left.

Can you insert a unicode character? On my machine, switch keyboard over to unicode, press Option (alt) and rattle onwards. To insert a special character control-command-spacebar gets fast to the character viewer and from there it is obvious. See.

This has implications for improving passwords, does it not?  pɹoʍssɐdʎɯ is quite an improvement.

Once you have discovered how to type unicode characters into a field (and learned the numbers) you could be quite significantly more clever in use of special characters for passwords: pɹoʍssɐd,   ₚₐₛₛᵩₒᴙᶛ   ,  ᴾᴬᵝᵂᴼᴿᴰ ,  bɿowƨƨɒq ,  ᶈᶏᶊᶊᶙᶕᶉᶁ ,  ⓟⓐⓢⓢ⒲⒪⒭⒟ and so on....

Musical symbols are found at 1D100, though most of the space is left, I think, as 'available'. I continue to think that musical notation could be very much extended for descriptions of intonation.

related pages 

251 unicode fun             197 hyphens and dashes