Another Angle: Solution

There are 12 regular (mostly easy) clues and 12 cryptic-type wordplay clues, each given in order of answer length (1 to 12). The answers are below:

• The highest grade (a)
• A type of Hawaiian lava (aa)
• Animal or baseball equipment (bat)
• Inverse of cosecant (sine)
• The claw on the foot of a bird of prey (talon)
• The point furthest from the Earth on an orbit around it (apogee)
• American (not British) name for a red insect with black spots (ladybug)
• Device for making sound from a CD (cdplayer)
• One who makes or otherwise works with locks (locksmith)
• Something to add to liquid to stop it from freezing, as its name suggests (antifreeze)
• An animal that eats grass (graminivore)
• Welcoming someone to their new house, or, used literally, turning on the heat inside (housewarming)

• First zone (first of zone = z)
• Jazz without German approval (jazz - zz = zz)
• Endless park (park without last letter = par)
• Overturned gas evaporates first (reversed gas + e = sage)
• Turned around, Japanese coin mantra (reversed (yen + om) = money)
• Overturned car in, say, ageless hole? ((reversed car) in ((say ageless = h-less) hole = ole) oracle)
• In sigh, first technique (in + sigh + first of technique = insight)
• Two hundred and one in Yemen port city die from diet (cci in aden + (diet - die) = accident)
• Before first road, California initially stole trains (first of before + road + ca + first letters of "stole trains" = broadcast)
• G8, say, keep Eritrea (g + ate + keep + er = gatekeeper)
• Six monster lizards containing smashed tin start mining (vi + gilas surrounding (anagrammed tin) + first of mining = vigilantism)
• United Nations and Germany train in red (un + de + train in red = undertrained)

Given the "left" and "right", there's only one way to put the answers in the grid:

Now, how do we get a letter from each row? The title of the puzzle suggests rotating something. One might try rotating the grid before filling it, but unfortunately then the corners aren't as given. Indeed, this is why the corners of the grid start out filled; they're not that helpful in actually solving the clues. It turns out what works is rotating the non-filled-in grid and seeing what letters in the filled-in-grid fall on the black spaces. Another thing solvers might try to do, which is roughly equally reasonable, is rotate the filled-in grid and see what letters fall on the black squares in the (unrotated) non-filled-in grid. (This is equivalent to entering the answers into the grid in a "rotated" way.) This will give an anagram of the 13 letters that the intended extraction method will give; in the case of a 180-degree rotation, this anagram will be just a reversal, which will hopefully allow solvers who try rotating the filled-in grid to see that they need to instead rotate the non-filled-in grid.

Following is the above grid with the letters taken from each row highlighted in green. Rotating the non-filled-in grid 90 degrees clockwise, we get:

Rotating 180 degrees (clockwise or counterclockwise), we get:

Finally, rotating 90 degrees counterclockwise, we get:

Combining all of these into one grid (and shading each differently), we get:

As seen above, rotating 90 degrees clockwise, we get "venusthismars", or "venus this mars". Rotating 180 degrees (clockwise or counterclockwise), we get "icewhenmelted", or "ice when melted". Rotating 90 degrees counterclockwise, we get "materialinsky", or "material in sky". These clue earth, water, and air (air somewhat ambiguously but far less so given the other clues), respectively, which are all classical elements. The answer is the fourth classical element, as clued by "The answer to this puzzle is missing" in the flavor text.

Answer: fire

Note: First I should say that the crossword grid CSS was taken from the solution to the MIT Mystery Hunt puzzle Clueless in 2018. Also, about "ideotype", it's obscure but it still has more hits than babewynery, though to be fair most of these hits aren't with the given definition and one might suspect the given definition arose from citogenesis. The definition is reworded enough that searching the given definition doesn't help. Probably most people won't get this word (unless they use a wordlist to backsolve it) but the puzzle can be solved missing a few words. It's probably possible to construct this puzzle without a word as obscure as "ideotype", but due to three letters being required from each row, it was too hard for me. I'm also somewhat sad about not being able to somehow use the idea of filling in the grid differently (for example, reading down rather than across), but there's then no one obvious way to extract letters and adding another three clue phrases would obviously be too constrained.

Note added on December 31, 2020, for replacement of "A specimen not from a taxon's type locality but still in the taxon according to the taxon author" with "Device for making sound from a disc": "ideotype" was just unfair in my opinion, as described above. I replaced it with something with the same required letters. Unfortunately, the replacement, "cdplayer", is two words. Hopefully this is not too confusing.

Note added on January 1, 2021: Replaced "disc" by "CD" in the aforemented clue, to put its difficulty more in line with that of the other clues.