The keys are sorted before processed by a reducer, using a
Raw comparator. The default comparator uses the compareTo method provided by the key type, which is a subclass of WritableComparable. Consider for example the following IntPair type:

public static class IntPair implements WritableComparable<IntPair> {
  private int first = 0;
  private int second = 0;
 
  public void set(int left, int right) { first = left; second = right; }
  public int getFirst() { return first; }
  public int getSecond() { return second; }
 
  public void readFields(DataInput in) throws IOException {
    first = in.readInt();
    second = in.readInt();
  }
  public void write(DataOutput out) throws IOException {
    out.writeInt(first);
    out.writeInt(second);
  }
 
  public int compareTo(IntPair o) {
    if (first != o.first) return first < o.first ? -1 : 1;
    else return second < o.second ? -1 : second == o.second ? 0 : 1;
  }
}

If we would like in a Hadoop job to sort the IntPair using the first element only, we can provide a RawComparator and set it using job.setSortComparatorClass:

public static class IntPair implements WritableComparable<IntPair> {
  ...
  public static class FirstOnlyComparator implements RawComparator<IntPair> {
    public int compare(byte[] b1, int s1, int l1, byte[] b2, int s2, int l2) {
      int first1 = WritableComparator.readInt(b1, s1);
      int first2 = WritableComparator.readInt(b2, s2);
      return first1 < first2 ? -1 : first1 == first2 ? 0 : 1;
    }
    public int compare(IntPair x, IntPair y) {
      return x.getFirst() < y.getFirst() ? -1 : x.getFirst() == y.getFirst() ? 0 : 1;
    }
  }
}
 
...
 
job.setSortComparatorClass(IntPair.FirstOnlyComparator.class);

Notice we used helper function readInt from WritableComparator class, which provides means of parsing primitive data types from byte streams.

In a reduce, it is guaranteed that keys are processed in ascending order. Sometimes it would be useful if the values associated with one key could also be processed in ascending order.

That is possible only to some degree. The (key, value) pairs are compared using the key only. After the (key, value) pairs are sorted, the (key, value) pairs with the same key are grouped together. This grouping can be performed using a custom RawComparator – it is therefore possible to group the input pairs using only a part of the keys.