The TL071 is not designed to be used with an input any less negative than 4V above its negative rail.
When powered from +/-15V, the input common mode voltage (from the datasheet) is between -12V and +15V.
In reality, you will probably get away with 2 diode drops above the negative power rail (about 1.4V - there is a huge clue when you see multiples of about 0.6 to 0.7V).
The reason for that is quite clear from the functional block doagram in section 8.2
The saturation you are seeing is due to phase inversion; this is a common issue with JFET input devices.
Most bipolar amplifiers will have a common mode range to the negative power rail (but only up to about V+ -1.4V); you could alternately look for a rail to rail input / output amplifier.
Some possible amplifiers:
LTC2057. Vcm V- to V+ - 1.5V
There are numerous offerings from TI, Maxim and ADI.