Javascript 470 453 438 434 432 421 404

s=String;s.prototype.a=s.prototype.replace;var a=prompt()/11.25,a=a+0.5|0,b,k,c=a,d=c%8,c=c/8|0,e=["north","east","south","west"],f,g,h;f=e[c];g=e[(c+1)%4];h=f==e[0]|f==e[2]?f+g:g+f;b="1;1 by 2;1-C;C by 1;C;C by 2;2-C;2 by 1".split(";")[d].a(1,f).a(2,g).a("C",h);k=b.a(/north/g,"N").a(/east/g,"E").a(/south/g,"S").a(/west/g,"W").a(/by/g,"b").a(/[\s-]/g,"");b=b[0].toUpperCase()+b.slice(1);alert(b+" "+k) 

You can copy this code to your console and execute it. It prompts you for the degrees input and outputs the result with alert();

Ungolfed Javascript can be found at this fiddle: http://jsfiddle.net/AezL3/11

Haskell 415 372 347 330 317 304 301C

Ended up converging on a solution like @VadimR's (and the symmetry's back!). Usage: h 219 outputs "Southwest by south SWbS"

d"N"="north" d"S"="south" d"E"="east" d"W"="west" d"b"=" by " d"-"="-" d(x:y)=d[x]++d y e(x:y)=x:tail(d$x:y) k 'N'='S' k 'S'='N' k 'E'='W' k x=x r="N NbE N-NE NEbN NE NEbE E-NE EbN E EbS E-SE SEbE SE SEbS S-SE SbE " p=words$r++(map k r) g x=p!!mod(round$x/11.25)32 h x=e(g x)++(filter(/='-')$' ':g x) 

3 more chars gone, thanks @shiona.

Python 3.8, 482 438 424 bytes

lambda h:' '.join([b(h),a(a(a(b(h)),1),d={' by ':'b','-':''})]) L=str.lower c={'North':'N','East':'E','South':'S','West':'W'} a=lambda t,l=0,d=c:[*(t:=t.replace([i,L(i)][l],d[i])for i in[*d])][-1] b=lambda h,k=[*c]:a('W|W by x|W-z|Z by w|Z|Z by x|X-z|X by w'.split('|')[int((q:=h*4/45+.5)%8)],d={'W':(W:=[*k][(v:=int(q//8)%4)]),'X':(X:=[*k][(v+1)%4]),'w':(w:=L(W)),'x':(x:=L(X)),'Z':(Z:=[W+x,X+w][W in'EastWest']),'z':L(Z)}) 

Try it online!

This is what I got after some golfing of tony goodwin's answer; posted at its own answer due to the TIO link being too long for a comment. If he chooses to update his answer to the above, I will delete this answer.

I am assuming that it's acceptable to submit a function as a solution rather than a full program. If not, here is a full program at 426 bytes.

I expect that much can still be done to shorten b.

Edit: Golfed off 44 bytes, courtesy of the majestic walrus. Still don't feel like b is done being golfed.

Edit2: Shaved off another 14 by unpacking dicts instead of using keys() and items().

Python, 2103 1647 1103 1034 924 889 848 bytes

Very late, I know. Thanks for the challenge, I am setting up a magnetometer for wind direction with my Pi, and wanted a 16 point of compass solution like this to feed into weather forecasting algorithms. All my code is in Python, so here is version of the javascript solution already posted in Python, but with an extra twist that you can specify either 32, 16, or 8 points of the compass at variable j, and I have changed the offset of degHead in the statement before it, depending on the number of points. I used a modified rename alogorithm (and used variables I could rename without corrupting the words!) to ensure I met the case requirements of the question.

I know this won't win, as Python is more wordy, and so am I.

Short version:

 def a(t,d,l): for i,j in d.items(): if l: i=i.lower() t=t.replace(i,j) return t def b(h,q): p=32 r=360 h=(h+(r/q/2))/(r/p) j=int(int(int(h %8)%8/(p/q))*p/q) h=int(h/8)%4 k=c.keys() u=['W','W by x','W-z','Z by w','Z','Z by x','X-z','X by w'] d={} d['W']=list(k)[h] d['w']=d['W'].lower() d['X']=list(k)[(h+1)%4] d['x']=d['X'].lower() if(d['W']=='North' or d['W']=='South'): d['Z']=d['W']+d['x'] else: d['Z']=d['X']+d['w'] d['z']=d['Z'].lower() return a(u[j],d,0) def g(n): n=a(n,c,0) n=a(n,c,1) d={'by':'b',' ':'','-':''} return a(n,d,0) def v(m): while True: try: return float(input(m)) except ValueError: print("?") c={'North':'N','East':'E','South':'S','West':'W'} while True: h=v("?") n=b(h,32) print(h,n,g(n)) 

Clear version

 import math import sys def calcPoint(degHead, points): maxPoints=32 if points not in(8,16,32): sys.exit("not a good question") degHead=(degHead+(360/points/2))/(360/maxPoints) j =int(int( int(degHead % 8)%8/(maxPoints/points))*maxPoints/points) degHead = int(degHead / 8) % 4 cardinal = ['North', 'East', 'South', 'West'] pointDesc = ['W', 'W by x', 'W-z', 'Z by w', 'Z', 'Z by x', 'X-z', 'X by w']#vars not compass points W = cardinal[degHead] X = cardinal[(degHead + 1) % 4] w=W.lower() x=X.lower() if (W == cardinal[0] or W == cardinal[2]) : Z =W + x else: Z =X + w z=Z.lower() return pointDesc[j].replace('W', W).replace('X', X).replace('w', w).replace('x', x).replace('Z', Z).replace('z', z); def getShortName(name): return name.replace('North', 'N').replace('East', 'E').replace('South', 'S').replace('West', 'W').replace('north', 'N').replace('east', 'E').replace('south', 'S').replace('west', 'W').replace('by', 'b').replace(' ', '').replace('-', '') def input_number(msg, err_msg=None): while True: try: return float(input(msg)) except ValueError: sys.exit("not a number") while True: headingCalib=input_number("input a number: ") print (headingCalib, end=' ') name = calcPoint(headingCalib,32) #degrees heading, points of compass 8,16 or 32) print (name, end=' ') shortName = getShortName(name) print (shortName)