Why am I getting an ORA-01722 (invalid number)?

When you say you checked the parameters do you mean the Parameters collection on the SqlCommand class? You might be falling foul of this note on the SqlParameter page:

Use caution when you use this overload of the SqlParameter constructor to specify integer parameter values. Because this overload takes a value of type Object, you must convert the integral value to an Object type when the value is zero, as the following C# example demonstrates. Copy

Parameter = new SqlParameter("@pname", Convert.ToInt32(0));

If you do not perform this conversion, the compiler assumes that you are trying to call the SqlParameter (string, SqlDbType) constructor overload.

I'd suggest you use something like

cmd.Parameters.Add(
   new SqlParameter("Field1", SqlDbType.Int32) { Value = field1Val });

instead to explicitly set the type.

command.BindByName = true;

Please see this explination.

I have given answer credit already, but I think it's worth mentioning here exactly what the root of my problems was, in case anyone else finds this item while looking for an answer to their own problem.

The problem is that the C# implementation of parameterized queries for Oracle contains a serious and potentially dangerous bug - a real "pit in the public domain":

It doesn't matter what you name your parameters; they have to be added in the order in which they appear in the query.

See more here.