**Spoiler: My answer**

Let's call the lines l

At any point in time, can define Turbo the Snail's current segment with a n-tuple: if l

Whenever Turbo turns left from l

However, between a segment on l

Turbo the Snail never visits the same segment both ways.

_{1}, ... , l_{n}, and give an orientation to each line.At any point in time, can define Turbo the Snail's current segment with a n-tuple: if l

_{i}is the line she's currently on, its i^{th}value is 1 (resp. -1) if she's moving in the line's positive (resp. negative) direction; otherwise, it's 1 (resp. -1) if she's on the left (resp. right) side of l_{i}from the point of view of someone traveling along it in the positive direction.Whenever Turbo turns left from l

_{i}to l_{j}, no value in the n-tuple changes - she doesn't cross, enter or leave any line other than l_{i}and l_{j}, and the n-tuple's definition ensures that the i^{th}and j^{th}values don't change. Similarly, when she turns right from l_{i}to l_{j}, the i^{th}and j^{th}are the only values that are changed. As a consequence of this, the product of the n values of the n-tuple is constant throughout Turbo's entire journey.However, between a segment on l

_{i}being passed both ways, the only variation in the n-tuple is the i^{th}value being flipped, which means that those two types of movement have opposite values for the product. This means it's impossible to access them both in the same travel.Turbo the Snail never visits the same segment both ways.