What is the True Pinout and Function of the 74ls181 in both of its modes (Active High and Active Low)?
Polarity of operands is a sort of arbitrary thing, a matter of data interpretation. What the different symbols mean is that if the data are treated as "active LOW", then the propagate signals also must be treated as "active LOW", and vice versa.

From the datasheet, page 5-2:


  As indicated, the LS181 can be used with either active LOW inputs 
  producing active LOW outputs or with active HIGH inputs 
  producing active HIGH outputs.


How does one go about sorting out this mess? This circuit was probably used last time about 30 years ago. The mess is usually sorted out by implementing a clean implementation of the ALU in FPGA using clear Verilog altogether with all control circuitry, pipelines, caches, microcode, and peripheral ports, all in chip much smaller than the LS181 itself.