The power of thinking
Your question looks like a "circuit puzzle" whose purpose is to make us think (a rare activity in this consumer age:) If you will allow me, I would use it in classes with my students to make them rack their brains. Okay, let's try to answer this challenge (I will use resources from my Circuit idea wikibook).
We can jokingly call this circuit an "inverting non-inverting amplifier" because it inverts the input voltage but has the properties of a non-inverting amplifier (high input resistance, common-mode signal, gain one unit greater than R2/R1). Let's see why…
The philosophy of negative feedback circuits
We can figuratively name this idea "active copying" because we make an (output) quantity Y equal to another (input) quantity X by subtracting them and adjusting Y so that to zero the difference (X - Y = 0). As a result, Y = X.
So, to implement this idea, we need a subtractor. We can make it by inverting one of the input quantities of a summer; thus we obtain a summer acting as a subtractor. But how do we sum voltages?
Kirchhoff's two laws give us an idea how to make two types of summers, respectively two types of negative feedback circuits - serial and parallel.
Series non-inverting amplifier (KVL). To make a series subtractor, we connect the two voltages (input and output) in series with grounded middle point. Note that they are connected in opposite direction when travelling the loop but in the same direction (polarity) regarding the ground. The result of subtraction is a floating voltage that needs a differential input. To ensure stability (negative feedback), we connect the output voltage to the inverting input and the input voltage to the non-inverting input. As a result, the circuit acts as a non-inverting amplifier with well-known properties - high input resistance, common-mode signal, differential input, gain 1 + R2/R1.
Parallel inverting amplifier (KCL). To make a parallel subtractor, we connect the two voltages (input and output) in "parallel" through resistors. Now they are connected in the same direction when travelling the loop but in the opposite direction (polarity) regarding the ground. In this case, the result of subtraction is the voltage of the middle point between the resistors (referenced to ground) that needs a humble single-ended input (the unused op-amp input is grounded). To ensure stability (negative feedback), this voltage should be connected to the inverting input. The input voltage changes this voltage in the same durection; so the circuit acts as an inverting amplifier with well-known properties - relatively low input resistance, single-ended input, gain -R2/R1.
These were the classic op-amp amplifier circuits. Let's now consider the "unconventional" OP configurations.
The classic op-amp circuits above were implemented by passive resistive feedback networks that are noninverting. If we replace them by inverting active networks, we must swap the op-amp inputs to keep the feedback negative. Thus we will obtain two more "unconventional" configurations with "odd" properties:
Series inverting amplifier (KVL) - high input resistance, common-mode signal, differential input, gain -(1 + R2/R1).
Parallel non-inverting amplifier (KCL) - relatively low input resistance, single-ended input, gain R2/R1.
See also my Wikibooks story about the related lab with my students in 2008.