Separating positive and negative terms in an algebraic expression

Question

I want to isolate positive and negative terms in this algebraic expression, for example. I have looked at the FullForm and tried many functions like Gatherby, Split etc. but they work on lists and the -1 is imbedded in a function Rational. From the Stack Exchange questions, I found that the best result is with:

expression/._?Positive->0

expression/._?Negative->0

But it adds few terms and misses others... Any quick help? The expression is:

expression = (a Nb^2)/2 + c^2 f$K^2 my$i$Re mz$i$Im Cos[θ]^2 + 
1/2 I bN$f c f$K my$i$Im Cos[θ] Sin[θ] - 
1/2 I bN$i c f$K my$i$Im Cos[θ] Sin[θ] + 
1/2 bN$f c f$K my$i$Re Cos[θ] Sin[θ] + 
1/2 bN$i c f$K my$i$Re Cos[θ] Sin[θ] + 
c^2 f$K^2 mx$i$Re mz$i$Im Cos[θ] Sin[θ] + 
1/2 I bN$f c f$K mx$i$Im Sin[θ]^2 - 
1/2 I bN$i c f$K mx$i$Im Sin[θ]^2 + 
1/2 bN$f c f$K mx$i$Re Sin[θ]^2 + 
1/2 bN$i c f$K mx$i$Re Sin[θ]^2 + 
1/2 c^2 f$K^2 my$i$Im^2 Cos[θ]^2 Sin[θ]^2 + 
1/2 c^2 f$K^2 my$i$Re^2 Cos[θ]^2 Sin[θ]^2 + 
c^2 f$K^2 mx$i$Im my$i$Im Cos[θ] Sin[θ]^3 + 
c^2 f$K^2 mx$i$Re my$i$Re Cos[θ] Sin[θ]^3 + 
1/2 c^2 f$K^2 mx$i$Im^2 Sin[θ]^4 + 
1/2 c^2 f$K^2 mx$i$Re^2 Sin[θ]^4

Answer 1

expression /. _?Internal`SyntacticNegativeQ :> 0

(a Nb^2)/2 + c^2 f$K^2 my$i$Re mz$i$Im Cos[θ]^2 + 
  1/2 I bN$f c f$K my$i$Im Cos[θ] Sin[θ] + 
  1/2 bN$f c f$K my$i$Re Cos[θ] Sin[θ] + 
  1/2 bN$i c f$K my$i$Re Cos[θ] Sin[θ] + 
  c^2 f$K^2 mx$i$Re mz$i$Im Cos[θ] Sin[θ] + 
  1/2 I bN$f c f$K mx$i$Im Sin[θ]^2 + 
  1/2 bN$f c f$K mx$i$Re Sin[θ]^2 + 
  1/2 bN$i c f$K mx$i$Re Sin[θ]^2 + 
  1/2 c^2 f$K^2 my$i$Im^2 Cos[θ]^2 Sin[θ]^2 + 
  1/2 c^2 f$K^2 my$i$Re^2 Cos[θ]^2 Sin[θ]^2 + 
  c^2 f$K^2 mx$i$Im my$i$Im Cos[θ] Sin[θ]^3 + 
  c^2 f$K^2 mx$i$Re my$i$Re Cos[θ] Sin[θ]^3 + 
  1/2 c^2 f$K^2 mx$i$Im^2 Sin[θ]^4 + 
  1/2 c^2 f$K^2 mx$i$Re^2 Sin[θ]^4

To replace the terms with a positive sign with 0:

Replace[expression , Except[_?Internal`SyntacticNegativeQ] :> 0, 1]

 -(1/2) I bN$i c f$K my$i$Im Cos[θ] Sin[θ] - 1/2 I bN$i c f$K mx$i$Im Sin[θ]^2