LSB becomes USB if we change sign of modulation sine?

Assume we have quadrature modulator:
a(t)=Cos(lo*t)*Cos(if*t)
b(t)=Cos(lo*t-Pi/2)*Cos(if*t-Pi/2)
with power combiner giving RF(t)=a(t)+b(t)
t - time
a(t) - output from mixing diode 1
b(t) - output from mixing diode 2 (with -pi/2 shifts on LO and IF)
lo - local oscillator frequency
if - modulating frequency
RF(t) - output from power combiner (combines a and b)

If frequency lo=100 Hz and if=5 Hz, then modulated output at RF will be 95 Hz (wich is LSB)

If we change modulating sine parameter to if=-5 Hz, then modulated output at rf will be 105Hz. Although it is SSB modulator with LSB output, changing sign in DSP formula will modulate output to upper side band.

Is it assumed that in SSB modulator input sine never can have negative frequency? We surely can use negative frequency, as it is audio data generated by software, and we can use any value put to sine.
Is it true, that using quadrature modulator we can generate signals both in LSB and USB by changing sign of modulation sinusoid?

Although it is SSB modulator with LSB output, changing sign in DSP formula will modulate output to upper side band.

Click to expand...

Why are you preceeding well-known signal processing facts with the word "although"? It's exactly as you assume.

My confusion comes from this: LSB modulator that uses quadrature mixer can generate signals in USB.