Are optimal turbine blades also optimal fan blades? In one case, the air drives the blade, in the other the blade drives the air.
Since each is 'just' the reverse of the other, one might expect that the aerodynamics are the same, with other variables being constant (e.g. if you powered a ducted turbine to drive fluid in the other direction, would another blade design be better than the one it already has).
 A: I'm not an expert either, but there are some arguments to think that they will often not be the same.
Blades are ideally aerofoil shaped, with a rounded front and a sharp trailing edge, so reversibility by having the rotor rotate in the opposite direction is not ideal as now the blades are oriented the wrong way. However the energy transfer direction can be reversed without changing the direction of spinning, by swapping the high and low pressure zones. In a turbine the fluid flows from high pressure to low pressure, in a fan the fan (in this case known as compressor or pump) pumps the fluid from low pressure to high pressure.
When the fluid is air, or any gas, and there is a substantial pressure change, I'm quite certain that the optimal turbine and optimal fan (or compressor) are not the same. Gas is compressible, so when it is compressed by a fan/compressor the volume of the gas shrinks. The fan/compressor blades of high performance compressors are shaped to take this into account, reducing the available cross sectional area between the blades smoothly as the gas flows through. A gas powered turbine is shaped in the reverse way, smoothly expanding the cross sectional area between the blades.
However fluids are known as non-compressible. Actually they are slightly compressible, but the amount of compression is so small that it can be ignored for most applications. I don't know if the compressibility of fluids is relevant in turbine/pump designs. If it is, this may result in small differences similar to compressible fluids, but maybe it can be ignored.
Another issue is that aerofoils are not necessarily symmetrical between the top and the bottom. This picture from Wikipedia (with a format that stackexchange doesn't support) shows some examples. While an aerofoil can be symmetric, usually the trailing edge is curved down, i.e. toward the higher pressure zone. I don't know when what type of aerofoil (or hydrofoil for fluids) is optimal in which circumstances, but likely the symmetrical shape isn't usually it. That means that - at least - the blade shape would need to be inverted to change an optimal pump into an optimal turbine.
It then depends on what you really want to know if inverting the blade shape counts as the same blade or not.
Having said that, pumped storage hydroelectric plants often use the same rotor both as pump and as turbine, so at least for that case the cost of having separate hardware apparently does not outweigh the loss of efficiency. However I don't know if these work by reversing the direction of rotation or if those have movable blades. (Hydro powerplants often have blades that can be rotated to adjust the flow volume and thus the power output without losing too much efficiency.)
A: I am not an expert in this, but I would suspect they would not be the same, related to Feynman's famous story.
The movements of the fluids are not the same in the two cases. 
I imagine a wall with the fan or turbine set in it. 
One way the fan sucks fluid in from one side, decreasing the pressure near the opening on that side of the wall, and pushes fluid out the other side. Since the moved fluid has inertia, it tends to move into the fluid on the other side.
The other way the fluid pressure on one side of the wall is higher than on the other side. High-pressure fluid turns your turbine, and lower-pressure fluid moves into the fluid on the other side.
The fan turns the same direction. One way it's high-pressure fluid in front of the fan and low-pressure fluid behind it, the other way it's the other way around. 
So I can imagine that this might make the aerodynamics not be the same. 
But maybe that doesn't make any difference.
Now take away the wall. The fan is stationary compared to the fluid all around it, and it moves some fluid. The fluid behind it moves relative to the stationary fluid around it, it has a velocity away from the fan.
The turbine is surrounded by fluid that has a velocity. It slows some of the fluid, and the fluid behind it moves slower than the fluid around that moving fluid. 
Those are not the 'reverse' of each other. So maybe the same blade design will not just get the same result in either case. 
I'd be interested to see an expert answer.
A: I think this is perhaps more of an Engineering question than a Physics one. In any case, the design of turbine and fan blades is quite different. I will assume the comparison here is between a fan and something like a wind turbine (rather than a high-temperature gas turbine, for example).
One of the main goals of an airfoil is to turn the flow as much as possible before separation takes place (which will cause a loss of efficiency). Airfoils are generally designed to be optimized for a particular flow velocity and angle of attack, which will cause separation to occur in a particular spot. Flow separation is a non-reversible phenomenon, so typically if a flow is reversed, separation will occur in a different place, with a differently-shaped velocity field. So, because of this, it is unlikely that an airfoil would be operating at peak efficiency in a reversed-flow situation.
