How to avoid if / else if chain when classifying a heading into 8 directions?

You can use map::lower_bound and store the upper-bound of each angle in a map.

Working example below:

#include <cassert>
#include <map>

enum Direction
{
    RIGHT,
    UP_RIGHT,
    UP,
    UP_LEFT,
    LEFT,
    DOWN_LEFT,
    DOWN,
    DOWN_RIGHT
};

using AngleDirMap = std::map<int, Direction>;

AngleDirMap map = {
    { 30, RIGHT },
    { 60, UP_RIGHT },
    { 120, UP },
    { 150, UP_LEFT },
    { 210, LEFT },
    { 240, DOWN_LEFT },
    { 300, DOWN },
    { 330, DOWN_RIGHT },
    { 360, RIGHT }
};

Direction direction(int angle)
{
    assert(angle >= 0 && angle <= 360);

    auto it = map.lower_bound(angle);
    return it->second;
}

int main()
{
    Direction d;

    d = direction(45);
    assert(d == UP_RIGHT);

    d = direction(30);
    assert(d == RIGHT);

    d = direction(360);
    assert(d == RIGHT);

    return 0;
}

#include <iostream>

enum Direction { UP, UP_RIGHT, RIGHT, DOWN_RIGHT, DOWN, DOWN_LEFT, LEFT, UP_LEFT };

Direction GetDirectionForAngle(int angle)
{
    const Direction slices[] = { RIGHT, UP_RIGHT, UP, UP, UP_LEFT, LEFT, LEFT, DOWN_LEFT, DOWN, DOWN, DOWN_RIGHT, RIGHT };
    return slices[(((angle % 360) + 360) % 360) / 30];
}

int main()
{
    // This is just a test case that covers all the possible directions
    for (int i = 15; i < 360; i += 30)
        std::cout << GetDirectionForAngle(i) << ' ';

    return 0;
}

This is how I would do it. (As per my previous comment).