Yes. Add a person at your customer's company as an external contact with the Project Collaborator role, and make them a member of the project. They can then comment on its tasks, reply by email, follow tasks and take on the ones you assign them.
Invite the customer's contact
In Contacts, open the Company Contacts tab and click Add Contact. If the person is already there, click their name instead.

Under Assign Roles, tick Project Collaborator. It's the minimum role for opening a project, and it doesn't use a licence.
Choose the project in Add to specific projects, or on the Project Membership tab for an existing contact. Administrators, Finance Managers and Portfolio Managers see these.
Leave Send Email Invite to User ticked and click Save.

Once the contact has the role, you can also add them from the project: open its settings, go to Members and click Add People. See project settings.
For other roles, such as approving timesheets or seeing invoices, see the customer portal.
Choose what they can do
On the project's Members tab, click the contact to open Person Details. Under Task Permissions, tick what they may do: Create, Update, Delete and Comment. Tick all four for full access, only Comment to let them discuss tasks without changing them, or none for view-only. People added with Add People start with all four.
Administrators and Finance Managers can open Project Settings on every project, Portfolio Managers on projects they're members of, and Project Managers on projects they're allowed to manage.
Work on tasks together
- Assign tasks: the project's external contacts are listed under External Contact in Assigned To. They're notified like your team. See task notifications.
- Follow: add them to a task's Followers so they hear about new comments and status changes. See task followers.
- Comment: they can post comments, @mention people and attach files, or reply to a task email to post a comment. See task discussions and replying to task emails.
- Add tasks: with Create, they can add tasks, or email them to the project's address. See the project email address.
- Chat: if they also have the Chat User role, they can discuss tasks from the Tasks tab in Chat. See chat for external contacts.
Task emails only reach people who've accepted their invitation. Each person chooses their notifications in notification settings.
Keep things within your team
- Team Only tasks are hidden from external contacts completely. See hiding tasks.
- Tick Post as Internal Note to hide one comment. A reply by email is never an internal note.
- The project's Project Notes are shown to everyone who opens the project, customer contacts included.
- With only Project Collaborator, a contact doesn't see time, costs or rates. With the Timesheet/Expense User role, which uses a licence, they can log time too.
Only add a contact to their own company's projects: they can open every project they're a member of.