Why did the manifold pressure gauge on the Ki-61 have negative values?
Disclaimer: I do not know if my answer is correct, but I think it is plausible.

The Ki-61 engine was the Kawasaki Ha-40, a Japanese version of the Daimler-Benz DB 601 Aa turbocharged inverted-V inline engine, so I would assume that the engine instrumentation was inspired by what was used in Germany at that time. Manifold pressure was measured in ata (Atmosphäre absolut, close to 1 bar), which is multiples of the standard atmospheric pressure. The picture below shows the manifold pressure gauge of the Me-109 G, which used the DB 605 A. This was derived from the DB 601, and the manifold pressure gauge has a range from 0.6 ata to 1.8 ata.


The DB 601 A had a maximum manifold pressure at sea level of 1.35 ata. This lets me assume that the Ha-40 was also turbocharged to 1.35 ata, which is 1.368 bar. The critical altitude was probably only 3700 m. The regular DB 601 A had a critical altitude of 4500 m, which was reduced in the DB 601 Aa export version to 3700 m. I could not find a direct reference for the critical altitude of the Ha-40.

At the 11,600 m service ceiling of the Ki-61, the atmospheric pressure is 206.6 mbar. Now we need to factor in the ram pressure gain of approx. 20% at 300 kts TAS, and with the 2.14 compression ratio needed for a critical altitude of 3700 m, this would translate to a manifold pressure of 530 mbar or 0.522 ata. If the Kawasaki engineers raised that to the 2.37 compression ratio of the original DB 601 A, the manifold pressure at 11,600 m could even had been 578 mbar or 0.570 ata.

The most plausible explanation for the numbers on the manifold pressure gauge now would be percentage over rsp. below 1 ata, the static pressure at sea level. Since it is more important to not overstress the engine at low level, the +40 upper limit fits well to the maximum 1.35 ata setting. This fits also well with the static value of 0, meaning 1 atmosphere. At high altitude the compressor would run at maximum setting anyway, and the scale here is for informing the pilot about available power, and would not cover the lowest values close to maximum altitude. 

I could not find a performance chart for the Ha-40 or the DB 601 A, and the nearest Google dug out for me is the performance chart of the DB 601 E below. Please note that this engine had more performance (2700 RPM and 1.42 ata manifold pressure for takeoff; the DB 601 A used 2400 RPM and 1.35 ata) and a higher critical altitude of 4800 m: