I have a game character swinging on a rope. I want him to crawl up when the mouse is above him and down when the mouse is below him.

But I want this to work from the character's point of view, which fails to work when he is swinging to the left because at PI radians you step immediately into -PI radians. This means it works fine doing < or > when swinging to the right but when he swings to the left the mouse always registers as above at -PI radians or less.

Example:
Code:

float theta;
float x,y;
void setup(){
  size(200, 200);
  smooth();
  strokeWeight(2);
  x = y = 100;
  theta = HALF_PI;
}
void draw(){
  background(255);
  stroke(0, 0, 255);
  line(0, y, x, y);
  float dx = mouseX - x;
  float dy = mouseY - y;
  float mTheta = atan2(dy, dx);
  if(mouseBelow(mTheta)){
    stroke(0,155,0);
  } 
  else {
    stroke(255,0,0);
  }
  line(x, y, mouseX, mouseY);
  // projected normals
  float vx = cos(theta) * 50;
  float vy = sin(theta) * 50;
  // right hand normal projected
  float rx = -vy;
  float ry = vx;
  // left hand normal projected
  float lx = vy;
  float ly = -vx;
  stroke(0);
  line(x + lx, y + ly, x + rx, y + ry);
  line(x, y, x + vx, y + vy);
  if(keyPressed){
    switch(keyCode){
    case LEFT:
	theta -= 0.1;
	break;
    case RIGHT:
	theta += 0.1;
	break;
    }
  }
}
boolean mouseBelow(float mTheta){
  float minTheta = theta - HALF_PI;
  float maxTheta = theta + HALF_PI;
  println("min:"+minTheta+" max:"+maxTheta+" theta:"+mTheta);
  return mTheta > minTheta && mTheta < maxTheta;
}
 

How can I get a relative angle check that overcomes the jump from +PI radians to -PI radians?

You could also just examine the sign of the dot product between your line's normal (towards the "inside" half space, that is: towards the "below" in your usage) and a vector constructed from the line to the mouse point.  If N.V==0 then on line, if <0 then outside or "above", if >0 then inside or "below".